#set document(title: "9.3 Use Properties of Angles, Triangles, and the Pythagorean Theorem", author: "OpenStax / XYZ Homework") #set page(width: 8.5in, height: auto, margin: 1in) #import "@preview/cetz:0.5.2" #set text(font: ("STIX Two Text", "Libertinus Serif", "New Computer Modern"), size: 10.5pt, lang: "en") #show math.equation: set text(font: ("STIX Two Math", "New Computer Modern Math")) #set par(justify: true, leading: 0.62em, spacing: 0.9em) #set enum(spacing: 1.1em) // room between list items so tall inline fractions don't collide #set list(spacing: 1.1em) #set table(stroke: 0.5pt + rgb("#c7ccd3")) #let BLUE = rgb("#183B6F") // brand navy — section bars + example/solution labels (white on navy 11.09:1) #let ORANGE = rgb("#A94509") // brand primary-700 — AA-safe deep orange for TEXT (5.93:1 on white; raw brand #F37021 is 2.94:1 and must never carry text) #let RED = rgb("#DC2626") // brand error-600 #let GREEN = rgb("#059669") // brand success-600 (decoration only; small green text uses green-text #007942) #show heading.where(level: 1): it => block(width: 100%, above: 0pt, below: 16pt, fill: gradient.linear(BLUE, rgb("#2C5AA0")), inset: (x: 14pt, y: 12pt), radius: 3pt, text(fill: white, weight: "bold", size: 19pt, it.body)) #show heading.where(level: 2): it => block(width: 100%, above: 18pt, below: 10pt, fill: BLUE, inset: (x: 10pt, y: 6pt), radius: 2pt, text(fill: white, weight: "bold", size: 12pt, it.body)) #show heading.where(level: 3): it => text(fill: ORANGE, weight: "bold", size: 12.5pt, it.body) #show heading.where(level: 4): it => text(fill: BLUE, weight: "bold", size: 10.5pt, it.body) #let examplebox(label, title, body) = block(width: 100%, breakable: true, fill: rgb("#EFF1F5"), stroke: 0.5pt + rgb("#CFDDF0"), radius: 4pt, inset: 10pt, above: 12pt, below: 12pt)[ #block(below: 6pt)[#box(fill: BLUE, inset: (x: 6pt, y: 2pt), radius: 2pt, text(fill: white, weight: "bold", size: 8.5pt, label)) #h(0.4em) #strong[#title]] #body] // rail = decorative left rule (raw brand token); labelcolor = AA-safe label text shade #let notebox(label, rail, labelcolor, tint, body) = block(width: 100%, breakable: true, fill: tint, stroke: (left: 3pt + rail), inset: (left: 10pt, rest: 8pt), radius: (right: 4pt), above: 11pt, below: 11pt)[ #text(fill: labelcolor, weight: "bold", size: 7.5pt, tracking: 0.5pt)[#upper(label)] #linebreak() #body] #let solutionbox(body) = block(above: 4pt, below: 8pt)[ #text(fill: BLUE, weight: "bold", size: 8.5pt)[Solution] #linebreak() #body] #let figph(msg) = block(width: 100%, height: 60pt, fill: rgb("#f6f7f9"), stroke: (paint: rgb("#c7ccd3"), dash: "dashed"), radius: 4pt, inset: 10pt)[ #align(center + horizon, text(fill: rgb("#889"), style: "italic", size: 9pt, msg))] // Standardize inlined figure sizes: measure the natural CeTZ canvas, then scale to a // consistent envelope (aspect-aware; see build_typst.py FIG_* constants). Unlike the // print preamble, dimensions are FLOORED: in an editor a user can trim a figure to a // degenerate 1-D shape (a bare line), and w/h or tw/w would then divide by zero. #let _STD_W = 3.5 #let _WIDE_W = 5.6 #let _MAX_H = 3.4 #let _ASPECT_WIDE = 2.2 #let _UPSCALE_MAX = 1.15 #let stdfig(body) = context { let m = measure(body) let w = calc.max(m.width / 1in, 0.01) let h = calc.max(m.height / 1in, 0.01) let tw = if w / h > _ASPECT_WIDE { _WIDE_W } else { _STD_W } let s = calc.min(tw / w, _MAX_H / h, _UPSCALE_MAX) align(center, box(scale(x: s * 100%, y: s * 100%, reflow: true, body))) } #show figure: set block(breakable: false) #set figure(gap: 8pt) #show figure.caption: set text(size: 8.5pt, fill: rgb("#555")) == 9.3#h(0.6em)Use Properties of Angles, Triangles, and the Pythagorean Theorem #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Before you get started, take this readiness quiz. Solve: #math.equation(block: false, alt: "x plus 3 plus 6 equals 11 .")[$x + 3 + 6 = 11 .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "x equals 2")[$x = 2$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Solve: #math.equation(block: false, alt: "the fraction a over 45 equals the fraction 4 over 3 .")[$frac(a, 45) = frac(4, 3) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "60")[$60$] ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Simplify: #math.equation(block: false, alt: "the square root of 36 plus 64 .")[$sqrt(36 + 64) .$] #linebreak() If you missed this problem, review . #solutionbox[ #math.equation(block: true, alt: "10")[$10$] ] ] So far in this chapter, we have focused on solving word problems, which are similar to many real-world applications of algebra. In the next few sections, we will apply our problem-solving strategies to some common geometry problems. === Use the Properties of Angles Are you familiar with the phrase ‘do a #math.equation(block: false, alt: "180 ’?")[$180 "’?"$] It means to turn so that you face the opposite direction. It comes from the fact that the measure of an angle that makes a straight line is #math.equation(block: false, alt: "180")[$180$] degrees. #figure(figph[The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees.], alt: "The image is a straight line with an arrow on each end. There is a dot in the center. There is an arrow pointing from one side of the dot to the other, and the angle is marked as 180 degrees.", caption: none) An #strong[angle] is formed by two rays that share a common endpoint. Each ray is called a side of the angle and the common endpoint is called the #strong[vertex]. An angle is named by its vertex. In , #math.equation(block: false, alt: "∠ A")[$"∠" A$] is the angle with vertex at point #math.equation(block: false, alt: "A .")[$A .$] The measure of #math.equation(block: false, alt: "∠ A")[$"∠" A$] is written #math.equation(block: false, alt: "m ∠ A .")[$m ∠ A .$] #figure(figph[The image is an angle made up of two rays. The angle is labeled with letter A.], alt: "The image is an angle made up of two rays. The angle is labeled with letter A.", caption: [#math.equation(block: false, alt: "∠ A")[$∠ A$] is the angle with vertex at #math.equation(block: false, alt: "point A .")[$"point" #h(0.2em) A .$]]) We measure angles in degrees, and use the symbol #math.equation(block: false, alt: "°")[$°$] to represent degrees. We use the abbreviation #math.equation(block: false, alt: "m")[$m$] for the #emph[measure] of an angle. So if #math.equation(block: false, alt: "∠ A")[$"∠" A$] is #math.equation(block: false, alt: "27° ,")[$"27°" ,$] we would write #math.equation(block: false, alt: "m ∠ A equals 27 .")[$m ∠ A = 27 .$] If the sum of the measures of two angles is #math.equation(block: false, alt: "180° ,")[$"180°" ,$] then they are called #strong[supplementary angles]. In , each pair of angles is supplementary because their measures add to #math.equation(block: false, alt: "180° .")[$"180°" .$] Each angle is the #emph[supplement] of the other. #figure(figph[Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.], alt: "Part a shows a 120 degree angle next to a 60 degree angle. Together, the angles form a straight line. Below the image, it reads 120 degrees plus 60 degrees equals 180 degrees. Part b shows a 45 degree angle attached to a 135 degree angle. Together, the angles form a straight line. Below the image, it reads 45 degrees plus 135 degrees equals 180 degrees.", caption: [The sum of the measures of supplementary angles is #math.equation(block: false, alt: "180° .")[$"180°" .$]]) If the sum of the measures of two angles is #math.equation(block: false, alt: "90° ,")[$"90°" ,$] then the angles are #strong[complementary angles]. In , each pair of angles is complementary, because their measures add to #math.equation(block: false, alt: "90° .")[$"90°" .$] Each angle is the #emph[complement] of the other. #figure(figph[Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.], alt: "Part a shows a 50 degree angle next to a 40 degree angle. Together, the angles form a right angle. Below the image, it reads 50 degrees plus 40 degrees equals 90 degrees. Part b shows a 60 degree angle attached to a 30 degree angle. Together, the angles form a right angle. Below the image, it reads 60 degrees plus 30 degrees equals 90 degrees.", caption: [The sum of the measures of complementary angles is #math.equation(block: false, alt: "90° .")[$"90°" .$]]) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Supplementary and Complementary Angles] If the sum of the measures of two angles is #math.equation(block: false, alt: "180° ,")[$"180°" ,$] then the angles are supplementary. If #math.equation(block: false, alt: "∠ A")[$"∠" A$] and #math.equation(block: false, alt: "∠ B")[$"∠" B$] are supplementary, then #math.equation(block: false, alt: "m ∠ A plus m ∠ B equals 180°.")[$m "∠" A + m "∠" B = "180°."$] If the sum of the measures of two angles is #math.equation(block: false, alt: "90° ,")[$"90°" ,$] then the angles are complementary. If #math.equation(block: false, alt: "∠ A")[$"∠" A$] and #math.equation(block: false, alt: "∠ B")[$"∠" B$] are complementary, then #math.equation(block: false, alt: "m ∠ A plus m ∠ B equals 90°.")[$m "∠" A + m "∠" B = "90°."$] ] In this section and the next, you will be introduced to some common geometry formulas. We will adapt our Problem Solving Strategy for Geometry Applications. The geometry formula will name the variables and give us the equation to solve. In addition, since these applications will all involve geometric shapes, it will be helpful to draw a figure and then label it with the information from the problem. We will include this step in the Problem Solving Strategy for Geometry Applications. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Use a Problem Solving Strategy for Geometry Applications.] + #strong[Read] the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information. + #emph[Identify] what you are looking for. + #emph[Name] what you are looking for and choose a variable to represent it. + #strong[Translate] into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. + #strong[Solve] the equation using good algebra techniques. + #strong[Check] the answer in the problem and make sure it makes sense. + #strong[Answer] the question with a complete sentence. ] The next example will show how you can use the Problem Solving Strategy for Geometry Applications to answer questions about supplementary and complementary angles. #examplebox("Example 1")[][ An angle measures #math.equation(block: false, alt: "40° .")[$"40°" .$] Find ⓐ its supplement, and ⓑ its complement. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓐ], []), [Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A geometry diagram shows a straight line with two angles marked. One angle, denoted by 's°', is an obtuse angle. The other angle, denoted by '40°', is an acute angle adjacent to the obtuse angle.], alt: "A geometry diagram shows a straight line with two angles marked. One angle, denoted by 's°', is an obtuse angle. The other angle, denoted by '40°', is an acute angle adjacent to the obtuse angle.", caption: none)], [Step 2. #strong[Identify] what you are looking for.], [#figure(figph[The text reads 'the supplement of a 40° angle.' against a white background.], alt: "The text reads 'the supplement of a 40° angle.' against a white background.", caption: none)], [Step 3. #strong[Name.] Choose a variable to represent it.], [#figure(figph[let s = the measure of the supplement], alt: "let s = the measure of the supplement", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula for the situation and substitute in the given information.], [#linebreak() #figure(figph[A mathematical equation, m∠A + m∠B = 180, illustrating that the sum of the measures of angles A and B is 180 degrees.], alt: "A mathematical equation, m∠A + m∠B = 180, illustrating that the sum of the measures of angles A and B is 180 degrees.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed on a white background: s + 40 = 180. This simple linear equation involves one variable 's', a constant, and an equals sign.], alt: "A mathematical equation is displayed on a white background: s + 40 = 180. This simple linear equation involves one variable 's', a constant, and an equals sign.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[The mathematical expression 's = 140' is displayed in black text against a plain white background.], alt: "The mathematical expression 's = 140' is displayed in black text against a plain white background.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[The equation '140 + 40 = 180' is displayed, with a question mark positioned directly above the equals sign, implying a query as to the truthfulness of the mathematical statement.], alt: "The equation '140 + 40 = 180' is displayed, with a question mark positioned directly above the equals sign, implying a query as to the truthfulness of the mathematical statement.", caption: none) #linebreak() #figure(figph[The image shows the equation '180 = 180' followed by a checkmark, indicating that the equality is correct or verified.], alt: "The image shows the equation '180 = 180' followed by a checkmark, indicating that the equality is correct or verified.", caption: none)], [], [Step 7. #strong[Answer] the question.], [#figure(figph[The image shows the text 'The supplement of the 40° angle is 140°.' This statement defines a supplementary angle relationship where the sum of two angles is 180 degrees.], alt: "The image shows the text 'The supplement of the 40° angle is 140°.' This statement defines a supplementary angle relationship where the sum of two angles is 180 degrees.", caption: none)], )) #figure(table( columns: 2, align: left, inset: 6pt, table.header([ⓑ], []), [Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A diagram with three rays originating from a common point, showing an angle labeled c degrees between the leftmost horizontal ray and a diagonal ray, and an angle of 40 degrees between the vertical ray and the diagonal ray.], alt: "A diagram with three rays originating from a common point, showing an angle labeled c degrees between the leftmost horizontal ray and a diagonal ray, and an angle of 40 degrees between the vertical ray and the diagonal ray.", caption: none)], [Step 2. #strong[Identify] what you are looking for.], [#figure(figph[The text reads, 'the complement of a 40° angle,' indicating a mathematical concept. The font is a sans-serif, dark teal color against a white background.], alt: "The text reads, 'the complement of a 40° angle,' indicating a mathematical concept. The font is a sans-serif, dark teal color against a white background.", caption: none)], [Step 3. #strong[Name.] Choose a variable to represent it.], [#figure(figph[let c = the measure of the complement], alt: "let c = the measure of the complement", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula for the situation and substitute in the given information.], [#linebreak() #figure(figph[A mathematical equation is displayed against a white background, reading 'm∠A + m∠B = 90'. This indicates that the measure of angle A plus the measure of angle B equals 90 degrees.], alt: "A mathematical equation is displayed against a white background, reading 'm∠A + m∠B = 90'. This indicates that the measure of angle A plus the measure of angle B equals 90 degrees.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A basic algebra equation is displayed, reading 'c + 40 = 90', suggesting a problem where one needs to solve for the variable 'c'.], alt: "A basic algebra equation is displayed, reading 'c + 40 = 90', suggesting a problem where one needs to solve for the variable 'c'.", caption: none) #linebreak() #figure(figph[The image shows a mathematical expression or variable assignment, 'c = 50', displayed in a clear, dark font against a plain white background.], alt: "The image shows a mathematical expression or variable assignment, 'c = 50', displayed in a clear, dark font against a plain white background.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A simple arithmetic problem, 50 + 40 = 90, is displayed with a question mark above the equals sign, asking if the statement is true. The equation is correct.], alt: "A simple arithmetic problem, 50 + 40 = 90, is displayed with a question mark above the equals sign, asking if the statement is true. The equation is correct.", caption: none) #linebreak() #figure(figph[The image displays the equation '90 = 90' followed by a checkmark, indicating that the equality is correct or verified.], alt: "The image displays the equation '90 = 90' followed by a checkmark, indicating that the equality is correct or verified.", caption: none)], [], [Step 7. #strong[Answer] the question.], [#figure(figph[The image shows text that reads: 'The complement of the 40° angle is 50°.'], alt: "The image shows text that reads: 'The complement of the 40° angle is 50°.'", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ An angle measures #math.equation(block: false, alt: "25° .")[$"25°" .$] Find its: ⓐ supplement ⓑ complement. #solutionbox[ + ⓐ 155° + ⓑ 65° ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ An angle measures #math.equation(block: false, alt: "77° .")[$"77°" .$] Find its: ⓐ supplement ⓑ complement. #solutionbox[ + ⓐ 103° + ⓑ 13° ] ] Did you notice that the words complementary and supplementary are in alphabetical order just like #math.equation(block: false, alt: "90")[$90$] and #math.equation(block: false, alt: "180")[$180$] are in numerical order? #examplebox("Example 2")[][ Two angles are supplementary. The larger angle is #math.equation(block: false, alt: "30°")[$"30°"$] more than the smaller angle. Find the measure of both angles. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A geometry diagram shows a straight line with two adjacent angles labeled 'a' and 'a + 30', where the line appears to be broken. The angles together form a straight line, implying they are supplementary.], alt: "A geometry diagram shows a straight line with two adjacent angles labeled 'a' and 'a + 30', where the line appears to be broken. The angles together form a straight line, implying they are supplementary.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [#figure(figph[The text in the image says 'the measures of both angles'.], alt: "The text in the image says 'the measures of both angles'.", caption: none)], [Step 3. #strong[Name.] Choose a variable to represent it. #linebreak() The larger angle is 30° more than the smaller angle.], [#figure(figph['let α = measure of smaller angle' in a dark teal font against a white background.], alt: "'let α = measure of smaller angle' in a dark teal font against a white background.", caption: none) #linebreak() #figure(figph[A mathematical equation states 'a + 30 = measure of larger angle'.], alt: "A mathematical equation states 'a + 30 = measure of larger angle'.", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula and substitute.], [#linebreak() #figure(figph[The image shows the mathematical equation m∠A + m∠B = 180, indicating that the sum of the measures of angle A and angle B is 180 degrees, meaning they are supplementary angles.], alt: "The image shows the mathematical equation m∠A + m∠B = 180, indicating that the sum of the measures of angle A and angle B is 180 degrees, meaning they are supplementary angles.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical equation is displayed with black text on a white background: (a + 30) + a = 180. This equation can be used to solve for the variable 'a'.], alt: "A mathematical equation is displayed with black text on a white background: (a + 30) + a = 180. This equation can be used to solve for the variable 'a'.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed: 2a + 30 = 180. This is a linear equation with one variable, 'a', that needs to be solved.], alt: "A mathematical equation is displayed: 2a + 30 = 180. This is a linear equation with one variable, 'a', that needs to be solved.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed on a white background, showing '2a = 150' in black font.], alt: "A mathematical equation is displayed on a white background, showing '2a = 150' in black font.", caption: none) #linebreak() #figure(figph[The text on a white background states 'a = 75 measure of smaller angle'.], alt: "The text on a white background states 'a = 75 measure of smaller angle'.", caption: none) #linebreak() #figure(figph[The image shows the expression 'a + 30 measure of larger angle' on a white background. The text describes a mathematical quantity for an angle.], alt: "The image shows the expression 'a + 30 measure of larger angle' on a white background. The text describes a mathematical quantity for an angle.", caption: none) #linebreak() #figure(figph[A mathematical expression displaying '75 + 30' on a white background.], alt: "A mathematical expression displaying '75 + 30' on a white background.", caption: none) #linebreak() #figure(figph[The number 105 is displayed in black text against a plain white background, centered in the frame.], alt: "The number 105 is displayed in black text against a plain white background, centered in the frame.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[The equation m∠A + m∠B = 180 is shown, indicating that the sum of the measures of angle A and angle B is 180 degrees. This implies that angles A and B are supplementary.], alt: "The equation m∠A + m∠B = 180 is shown, indicating that the sum of the measures of angle A and angle B is 180 degrees. This implies that angles A and B are supplementary.", caption: none) #linebreak() #figure(figph[A mathematical expression asks whether 75 + 105 equals 180, with a question mark placed above the equals sign.], alt: "A mathematical expression asks whether 75 + 105 equals 180, with a question mark placed above the equals sign.", caption: none) #linebreak() #figure(figph[The image displays the equation '180 = 180' followed by a checkmark, indicating that the equality is correct or verified.], alt: "The image displays the equation '180 = 180' followed by a checkmark, indicating that the equality is correct or verified.", caption: none)], [], [Step 7. #strong[Answer] the question.], [#figure(figph[The measures of the angles are 75° and 105°.], alt: "The measures of the angles are 75° and 105°.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Two angles are supplementary. The larger angle is #math.equation(block: false, alt: "100°")[$"100°"$] more than the smaller angle. Find the measures of both angles. #solutionbox[ 40°, 140° ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Two angles are complementary. The larger angle is #math.equation(block: false, alt: "40°")[$"40°"$] more than the smaller angle. Find the measures of both angles. #solutionbox[ 25°, 65° ] ] === Use the Properties of Triangles What do you already know about triangles? Triangle have three sides and three angles. Triangles are named by their vertices. The #strong[triangle] in is called #math.equation(block: false, alt: "Δ A B C ,")[$"Δ" A B C ,$] read ‘triangle #math.equation(block: false, alt: "ABC")[$"ABC"$]’. We label each side with a lower case letter to match the upper case letter of the opposite vertex. #figure(figph[The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.], alt: "The vertices of the triangle on the left are labeled A, B, and C. The sides are labeled a, b, and c.", caption: [#math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] has vertices #math.equation(block: false, alt: "A , B , and C")[$A , B , #h(0.2em) "and" #h(0.2em) C$] and sides #math.equation(block: false, alt: "a , b , and c .")[$a , b , "and" #h(0.2em) c "."$]]) The three angles of a triangle are related in a special way. The sum of their measures is #math.equation(block: false, alt: "180° .")[$"180°" .$] #math.equation(block: true, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180°")[$m "∠" A + m "∠" B + m "∠" C = "180°"$]#notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Sum of the Measures of the Angles of a Triangle] For any #math.equation(block: false, alt: "Δ A B C ,")[$"Δ" A B C ,$] the sum of the measures of the angles is #math.equation(block: false, alt: "180° .")[$"180°" .$] #math.equation(block: true, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180°")[$m "∠" A + m "∠" B + m "∠" C = "180°"$] ] #examplebox("Example 3")[][ The measures of two angles of a triangle are #math.equation(block: false, alt: "55°")[$"55°"$] and #math.equation(block: false, alt: "82° .")[$"82°" .$] Find the measure of the third angle. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[A triangle ABC is shown, with angle A measuring 82 degrees, angle B measuring 55 degrees, and angle C denoted by x.], alt: "A triangle ABC is shown, with angle A measuring 82 degrees, angle B measuring 55 degrees, and angle C denoted by x.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [#figure(figph[the measure of the third angle in a triangle], alt: "the measure of the third angle in a triangle", caption: none)], [Step 3. #strong[Name.] Choose a variable to represent it.], [#figure(figph[The image shows the text 'let x = the measure of the angle' in a sans-serif font, suggesting a mathematical definition or problem statement.], alt: "The image shows the text 'let x = the measure of the angle' in a sans-serif font, suggesting a mathematical definition or problem statement.", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula and substitute.], [#linebreak() #figure(figph[The image shows the angle sum property of a triangle, where the sum of the measures of interior angles A, B, and C is equal to 180 degrees, represented as m∠A + m∠B + m∠C = 180.], alt: "The image shows the angle sum property of a triangle, where the sum of the measures of interior angles A, B, and C is equal to 180 degrees, represented as m∠A + m∠B + m∠C = 180.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical equation is displayed on a white background: 55 + 82 + x = 180. This equation is commonly used to find the third angle of a triangle when two angles are known, as the sum of angles in a triangle is 180 degrees.], alt: "A mathematical equation is displayed on a white background: 55 + 82 + x = 180. This equation is commonly used to find the third angle of a triangle when two angles are known, as the sum of angles in a triangle is 180 degrees.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed, reading '137 + x = 180'.], alt: "A mathematical equation is displayed, reading '137 + x = 180'.", caption: none) #linebreak() #figure(figph[The mathematical equation 'x = 43' is displayed in the top right corner against a plain white background.], alt: "The mathematical equation 'x = 43' is displayed in the top right corner against a plain white background.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical equation showing 55 + 82 + 43 with a question mark above the equals sign, followed by 180, implying a verification of the sum equaling 180.], alt: "A mathematical equation showing 55 + 82 + 43 with a question mark above the equals sign, followed by 180, implying a verification of the sum equaling 180.", caption: none) #linebreak() #figure(figph[The mathematical equality 180 = 180 is displayed, followed by a checkmark indicating its correctness.], alt: "The mathematical equality 180 = 180 is displayed, followed by a checkmark indicating its correctness.", caption: none)], [], [Step 7. #strong[Answer] the question.], [#figure(figph[The measure of the third angle is 43 degrees.], alt: "The measure of the third angle is 43 degrees.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The measures of two angles of a triangle are #math.equation(block: false, alt: "31°")[$"31°"$] and #math.equation(block: false, alt: "128° .")[$"128°" .$] Find the measure of the third angle. #solutionbox[ 21° ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ A triangle has angles of #math.equation(block: false, alt: "49°")[$"49°"$] and #math.equation(block: false, alt: "75° .")[$"75°" .$] Find the measure of the third angle. #solutionbox[ 56° ] ] ==== Right Triangles Some triangles have special names. We will look first at the #strong[right triangle]. A right triangle has one #math.equation(block: false, alt: "90°")[$"90°"$] angle, which is often marked with the symbol shown. #figure(figph[A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.], alt: "A right triangle is shown. The right angle is marked with a box and labeled 90 degrees.", caption: none) If we know that a triangle is a right triangle, we know that one angle measures #math.equation(block: false, alt: "90°")[$"90°"$] so we only need the measure of one of the other angles in order to determine the measure of the third angle. #examplebox("Example 4")[][ One angle of a right triangle measures #math.equation(block: false, alt: "28° .")[$"28°" .$] What is the measure of the third angle? #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [#figure(figph[Geometric diagram of a right-angled triangle labeled ABC, showing angles 90 degrees at A, 28 degrees at B, and x degrees at C, illustrating a problem to find angle x.], alt: "Geometric diagram of a right-angled triangle labeled ABC, showing angles 90 degrees at A, 28 degrees at B, and x degrees at C, illustrating a problem to find angle x.", caption: none)]), [Step 2. #strong[Identify] what you are looking for.], [#figure(figph[The text reads, 'the measure of an angle'.], alt: "The text reads, 'the measure of an angle'.", caption: none)], [Step 3. #strong[Name.] Choose a variable to represent it.], [#figure(figph[The text reads 'let x = the measure of the angle' in a dark teal font on a white background, representing a mathematical definition of a variable x.], alt: "The text reads 'let x = the measure of the angle' in a dark teal font on a white background, representing a mathematical definition of a variable x.", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula and substitute.], [#linebreak() #figure(figph[The equation m angle A plus m angle B plus m angle C equals 180.], alt: "The equation m angle A plus m angle B plus m angle C equals 180.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical equation is displayed, showing 'x + 90 + 28 = 180' in black text against a white background.], alt: "A mathematical equation is displayed, showing 'x + 90 + 28 = 180' in black text against a white background.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed with dark gray text on a white background, reading 'x + 118 = 180'.], alt: "A mathematical equation is displayed with dark gray text on a white background, reading 'x + 118 = 180'.", caption: none) #linebreak() #figure(figph[The image displays a simple mathematical equation, 'x = 62', written in black text against a plain white background, asserting that the variable x has a value of 62.], alt: "The image displays a simple mathematical equation, 'x = 62', written in black text against a plain white background, asserting that the variable x has a value of 62.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical equation checks if 180 is equal to the sum of 90, 28, and 62, posing '180 =? 90 + 28 + 62' on a white background. The sum is 180, so the equality is true.], alt: "A mathematical equation checks if 180 is equal to the sum of 90, 28, and 62, posing '180 =? 90 + 28 + 62' on a white background. The sum is 180, so the equality is true.", caption: none) #linebreak() #figure(figph[The image displays the equation '180 = 180' with a checkmark, signifying a correct or verified mathematical statement.], alt: "The image displays the equation '180 = 180' with a checkmark, signifying a correct or verified mathematical statement.", caption: none)], [], [Step 7. #strong[Answer] the question.], [#figure(figph[The text states, 'The measure of the third angle is 62×0.'], alt: "The text states, 'The measure of the third angle is 62×0.'", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ One angle of a right triangle measures #math.equation(block: false, alt: "56° .")[$"56°" .$] What is the measure of the other angle? #solutionbox[ 34° ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ One angle of a right triangle measures #math.equation(block: false, alt: "45° .")[$"45°" .$] What is the measure of the other angle? #solutionbox[ 45° ] ] In the examples so far, we could draw a figure and label it directly after reading the problem. In the next example, we will have to define one angle in terms of another. So we will wait to draw the figure until we write expressions for all the angles we are looking for. #examplebox("Example 5")[][ The measure of one angle of a right triangle is #math.equation(block: false, alt: "20°")[$"20°"$] more than the measure of the smallest angle. Find the measures of all three angles. #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem.], []), [Step 2. #strong[Identify] what you are looking for.], [the measures of all three angles], [Step 3. #strong[Name.] Choose a variable to represent it. #linebreak() #linebreak() #linebreak() Now draw the figure and label it with the given information.], [#figure(figph[The text], alt: "The text", caption: none) #linebreak() #figure(figph[The image displays a mathematical equation written horizontally, 'a + 20 = 2nd angle', indicating a relationship where 'a plus 20' equals the second angle.], alt: "The image displays a mathematical equation written horizontally, 'a + 20 = 2nd angle', indicating a relationship where 'a plus 20' equals the second angle.", caption: none) #linebreak() #figure(figph[The image shows text that states '90 = 3rd angle (the right angle)', indicating that the third angle in a context, likely a geometric figure, is a right angle measuring 90 degrees.], alt: "The image shows text that states '90 = 3rd angle (the right angle)', indicating that the third angle in a context, likely a geometric figure, is a right angle measuring 90 degrees.", caption: none) #linebreak() #figure(figph[A right-angled triangle ABC with the right angle at C. Angle A is denoted as 'a' and angle B is denoted as 'a + 20'. The sum of angles in a triangle is 180 degrees.], alt: "A right-angled triangle ABC with the right angle at C. Angle A is denoted as 'a' and angle B is denoted as 'a + 20'. The sum of angles in a triangle is 180 degrees.", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula and substitute into the formula.], [#figure(figph[Equation: m∠A + m∠B + m∠C = 180, illustrating the angle sum property of a triangle.], alt: "Equation: m∠A + m∠B + m∠C = 180, illustrating the angle sum property of a triangle.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed, showing a + (a + 20) + 90 = 180, an algebraic expression typically used to solve for an unknown variable 'a' in geometry or algebra problems.], alt: "A mathematical equation is displayed, showing a + (a + 20) + 90 = 180, an algebraic expression typically used to solve for an unknown variable 'a' in geometry or algebra problems.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[A mathematical equation is displayed, reading '2a + 110 = 180' in black text against a white background.], alt: "A mathematical equation is displayed, reading '2a + 110 = 180' in black text against a white background.", caption: none) #linebreak() #figure(figph[A mathematical equation is displayed on a white background, reading '2a = 70'.], alt: "A mathematical equation is displayed on a white background, reading '2a = 70'.", caption: none) #linebreak() #figure(figph[The image displays the equation 'a = 35 first angle' in black and teal text on a white background, indicating a variable 'a' is equal to 35, referred to as the first angle.], alt: "The image displays the equation 'a = 35 first angle' in black and teal text on a white background, indicating a variable 'a' is equal to 35, referred to as the first angle.", caption: none) #linebreak() #figure(figph[The image shows the text 'a + 20 second angle' in a dark grey color on a white background.], alt: "The image shows the text 'a + 20 second angle' in a dark grey color on a white background.", caption: none) #linebreak() #figure(figph[A simple arithmetic expression is displayed on a white background, showing the addition problem '35 + 20.' The number 35 is rendered in red text, while the plus sign and the number 20 are in black.], alt: "A simple arithmetic expression is displayed on a white background, showing the addition problem '35 + 20.' The number 35 is rendered in red text, while the plus sign and the number 20 are in black.", caption: none) #linebreak() #figure(figph[The number '55' is displayed in a simple, clear black font on a plain white background.], alt: "The number '55' is displayed in a simple, clear black font on a plain white background.", caption: none) #linebreak() #figure(figph[The text '90 third angle' is displayed on a white background, indicating a numerical value and a descriptive term related to an angle or perspective.], alt: "The text '90 third angle' is displayed on a white background, indicating a numerical value and a descriptive term related to an angle or perspective.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A mathematical equation reads '35 + 55 + 90 ?= 180', questioning if the sum of 35, 55, and 90 equals 180.], alt: "A mathematical equation reads '35 + 55 + 90 ?= 180', questioning if the sum of 35, 55, and 90 equals 180.", caption: none) #linebreak() #figure(figph[The image displays the equation '180 = 180' followed by a checkmark, indicating that the equality is correct or verified.], alt: "The image displays the equation '180 = 180' followed by a checkmark, indicating that the equality is correct or verified.", caption: none)], [], [Step 7. #strong[Answer] the question.], [#figure(figph[The three angles measure 35°, 55°, and 90°.], alt: "The three angles measure 35°, 55°, and 90°.", caption: none)], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The measure of one angle of a right triangle is #math.equation(block: false, alt: "50°")[$"50°"$] more than the measure of the smallest angle. Find the measures of all three angles. #solutionbox[ 20°, 70°, 90° ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ The measure of one angle of a right triangle is #math.equation(block: false, alt: "30°")[$"30°"$] more than the measure of the smallest angle. Find the measures of all three angles. #solutionbox[ 30°, 60°, 90° ] ] ==== Similar Triangles When we use a map to plan a trip, a sketch to build a bookcase, or a pattern to sew a dress, we are working with similar figures. In geometry, if two figures have exactly the same shape but different sizes, we say they are #strong[similar figures]. One is a scale model of the other. The corresponding sides of the two figures have the same ratio, and all their corresponding angles have the same measures. The two triangles in are similar. Each side of #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is four times the length of the corresponding side of #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$] and their corresponding angles have equal measures. #figure(figph[Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.], alt: "Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 16, the side across from B is labeled 20, and the side across from C is labeled 12. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4, the side across from Y is labeled 5, and the side across from Z is labeled 3. Beside the triangles, it says that the measure of angle A equals the measure of angle X, the measure of angle B equals the measure of angle Y, and the measure of angle C equals the measure of angle Z. Below this is the proportion 16 over 4 equals 20 over 5 equals 12 over 3.", caption: [#math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] and #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$] are similar triangles. Their corresponding sides have the same ratio and the corresponding angles have the same measure.]) #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[Properties of Similar Triangles ] If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths are in the same ratio. #figure(figph[Two similar triangles are shown: the left larger triangle has its legs labeled lowercase c, a, b, with points labeled as uppercase A, B, and C. The right smaller triangle mirrors the left by having its legs labeled as lowercase z, x, y, and its points labeled X, Y, and Z. ], alt: "Two similar triangles are shown: the left larger triangle has its legs labeled lowercase c, a, b, with points labeled as uppercase A, B, and C. The right smaller triangle mirrors the left by having its legs labeled as lowercase z, x, y, and its points labeled X, Y, and Z.", caption: none) ] The length of a side of a triangle may be referred to by its endpoints, two vertices of the triangle. For example, in #math.equation(block: false, alt: "Δ A B C :")[$"Δ" A B C ":"$] #math.equation(block: true, alt: "the length a can also be written B C; the length b can also be written A C; the length c can also be written A B")[$#h(2em) "the length" #h(0.2em) a #h(0.2em) "can also be written" #h(0.2em) B C \ "the length" #h(0.2em) b #h(0.2em) "can also be written" #h(0.2em) A C \ "the length" #h(0.2em) c #h(0.2em) "can also be written" #h(0.2em) A B$] We will often use this notation when we solve similar triangles because it will help us match up the corresponding side lengths. #examplebox("Example 6")[][ #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] and #math.equation(block: false, alt: "Δ X Y Z")[$"Δ" X Y Z$] are similar triangles. The lengths of two sides of each triangle are shown. Find the lengths of the third side of each triangle. #figure(figph[Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.], alt: "Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 3.2, and the side across from C is labeled 4. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 4.5, the side across from Y is labeled y, and the side across from Z is labeled 3.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, [Step 1. #strong[Read] the problem. Draw the figure and label it with the given information.], [The figure is provided.], [Step 2. #strong[Identify] what you are looking for.], [The length of the sides of similar triangles], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #linebreak() #emph[a] = length of the third side of #math.equation(block: false, alt: "Δ A B C")[$Δ A B C$] #linebreak() #emph[y] = length of the third side #math.equation(block: false, alt: "Δ X Y Z")[$Δ X Y Z$]], [Step 4. #strong[Translate.]], [], [The triangles are similar, so the corresponding sides are in the same ratio. So #linebreak() #math.equation(block: false, alt: "the fraction A B over X Y equals the fraction B C over Y Z equals the fraction A C over X Z")[$frac(A B, X Y) = frac(B C, Y Z) = frac(A C, X Z)$] #linebreak() Since the side #math.equation(block: false, alt: "A B equals 4")[$A B = 4$] corresponds to the side #math.equation(block: false, alt: "X Y equals 3")[$X Y = 3$], we will use the ratio #math.equation(block: false, alt: "the fraction AB over XY equals the fraction 4 over 3")[$frac("AB", "XY") = frac(4, 3)$] to find the other sides. #linebreak() #linebreak() Be careful to match up corresponding sides correctly. #linebreak() #figure(figph[This image displays the steps to calculate unknown side lengths 'a' and 'y' in similar triangles. It shows the ratios of corresponding sides from large and small triangles, setting up proportions like 4/3 = a/4.5 and 4/3 = 3.2/y.], alt: "This image displays the steps to calculate unknown side lengths 'a' and 'y' in similar triangles. It shows the ratios of corresponding sides from large and small triangles, setting up proportions like 4/3 = a/4.5 and 4/3 = 3.2/y.", caption: none)], [], [Step 5. #strong[Solve] the equation.], [#figure(figph[Two sets of equations are solved, showing calculations for 'a' and 'y'. The first set solves 3a = 4(4.5) to find a=6. The second set solves 4y = 3(3.2) to find y=2.4.], alt: "Two sets of equations are solved, showing calculations for 'a' and 'y'. The first set solves 3a = 4(4.5) to find a=6. The second set solves 4y = 3(3.2) to find y=2.4.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[Two examples demonstrate how to check if two fractions are proportional by using the cross-multiplication method, showing the resulting equalities with a checkmark.], alt: "Two examples demonstrate how to check if two fractions are proportional by using the cross-multiplication method, showing the resulting equalities with a checkmark.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The third side of #math.equation(block: false, alt: "Δ A B C")[$Δ A B C$] is 6 and the third side of #math.equation(block: false, alt: "Δ X Y Z")[$Δ X Y Z$] is 2.4.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z .")[$"Δ" X Y Z .$] Find #math.equation(block: false, alt: "a .")[$a .$] #figure(figph[Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.], alt: "Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.", caption: none) #solutionbox[ 8 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z .")[$"Δ" X Y Z .$] Find #math.equation(block: false, alt: "y .")[$y .$] #figure(figph[Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.], alt: "Two triangles are shown. They appear to be the same shape, but the triangle on the right is larger The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled a, the side across from B is labeled 15, and the side across from C is labeled 17. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled 12, the side across from Y is labeled y, and the side across from Z is labeled 25.5.", caption: none) #solutionbox[ 22.5 ] ] === Use the Pythagorean Theorem The #strong[Pythagorean Theorem] is a special property of right triangles that has been used since ancient times. It is named after the Greek philosopher and mathematician Pythagoras who lived around #math.equation(block: false, alt: "500")[$500$] BCE. Remember that a right triangle has a #math.equation(block: false, alt: "90°")[$"90°"$] angle, which we usually mark with a small square in the corner. The side of the triangle opposite the #math.equation(block: false, alt: "90°")[$"90°"$] angle is called the #strong[hypotenuse], and the other two sides are called the #strong[legs]. #figure(figph[Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”], alt: "Three right triangles are shown. Each has a box representing the right angle. The first one has the right angle in the lower left corner, the next in the upper left corner, and the last one at the top. The two sides touching the right angle are labeled “leg” in each triangle. The sides across from the right angles are labeled “hypotenuse.”", caption: [In a right triangle, the side opposite the #math.equation(block: false, alt: "90°")[$"90°"$] angle is called the hypotenuse and each of the other sides is called a leg.]) The Pythagorean Theorem tells how the lengths of the three sides of a right triangle relate to each other. It states that in any right triangle, the sum of the squares of the two legs equals the square of the hypotenuse. #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[The Pythagorean Theorem] In any right triangle #math.equation(block: false, alt: "Δ A B C ,")[$"Δ" A B C ,$] #math.equation(block: true, alt: "a squared plus b squared equals c squared")[$a^(2) + b^(2) = c^(2)$]where #math.equation(block: false, alt: "c")[$c$] is the length of the hypotenuse #math.equation(block: false, alt: "a")[$a$] and #math.equation(block: false, alt: "b")[$b$] are the lengths of the legs. #figure(figph[A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.], alt: "A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked a and b.", caption: none) ] To solve problems that use the Pythagorean Theorem, we will need to find square roots. In Simplify and Use Square Roots we introduced the notation #math.equation(block: false, alt: "the square root of m")[$sqrt(m)$] and defined it in this way: #math.equation(block: true, alt: "If m equals n squared , then the square root of m equals n for n greater than or equal to 0")[$"If" #h(0.2em) m = n^(2) , #h(0.2em) "then" #h(0.2em) sqrt(m) = n #h(0.2em) "for" #h(0.2em) n ≥ 0$]For example, we found that #math.equation(block: false, alt: "the square root of 25")[$sqrt(25)$] is #math.equation(block: false, alt: "5")[$5$] because #math.equation(block: false, alt: "5 squared equals 25 .")[$5^(2) = 25 .$] We will use this definition of square roots to solve for the length of a side in a right triangle. #examplebox("Example 7")[][ Use the Pythagorean Theorem to find the length of the hypotenuse. #figure(figph[Right triangle with legs labeled as 3 and 4.], alt: "Right triangle with legs labeled as 3 and 4.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem.], []), [Step 2. #strong[Identify] what you are looking for.], [the length of the hypotenuse of the triangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "c equals the length of the hypotenuse")[$c = "the length of the hypotenuse"$] #linebreak() #figure(figph[A right-angled triangle with legs of length 3 and 4, and the hypotenuse labeled 'c'. This classic 3-4-5 triangle demonstrates the Pythagorean theorem.], alt: "A right-angled triangle with legs of length 3 and 4, and the hypotenuse labeled 'c'. This classic 3-4-5 triangle demonstrates the Pythagorean theorem.", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Illustration of the Pythagorean theorem (a² + b² = c²) and a specific example (3² + 4² = c²) used to find the hypotenuse.], alt: "Illustration of the Pythagorean theorem (a² + b² = c²) and a specific example (3² + 4² = c²) used to find the hypotenuse.", caption: none)], [Step 5. #strong[Solve] the equation.], [#figure(figph[Solving for 'c': The image displays the calculation starting from 9 + 16 = c^2, simplifying to 25 = c^2, and then taking the square root to determine that c = 5.], alt: "Solving for 'c': The image displays the calculation starting from 9 + 16 = c^2, simplifying to 25 = c^2, and then taking the square root to determine that c = 5.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[This image demonstrates the Pythagorean triple (3,4,5) with a step-by-step verification that 3^2 + 4^2 = 5^2, resulting in 9 + 16 = 25 and finally 25 = 25.], alt: "This image demonstrates the Pythagorean triple (3,4,5) with a step-by-step verification that 3^2 + 4^2 = 5^2, resulting in 9 + 16 = 25 and finally 25 = 25.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The length of the hypotenuse is 5.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the Pythagorean Theorem to find the length of the hypotenuse. #figure(figph[A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.], alt: "A right triangle is shown. The right angle is marked with a box. Across from the box is side c. The sides touching the right angle are marked 6 and 8.", caption: none) #solutionbox[ 10 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the Pythagorean Theorem to find the length of the hypotenuse. #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as c. One of the sides touching the right angle is labeled as 15, the other is labeled “8”.", caption: none) #solutionbox[ 17 ] ] #examplebox("Example 8")[][ Use the Pythagorean Theorem to find the length of the longer leg. #figure(figph[Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.], alt: "Right triangle is shown with one leg labeled as 5 and hypotenuse labeled as 13.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem.], []), [Step 2. #strong[Identify] what you are looking for.], [The length of the leg of the triangle], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #math.equation(block: false, alt: "b equals the leg of the triangle")[$b = "the leg of the triangle"$] #linebreak() Label side #emph[b] #linebreak() #figure(figph[A right-angled triangle with one vertical leg of length 5, a horizontal leg labeled 'b', and the hypotenuse of length 13. A square symbol indicates the right angle between the legs.], alt: "A right-angled triangle with one vertical leg of length 5, a horizontal leg labeled 'b', and the hypotenuse of length 13. A square symbol indicates the right angle between the legs.", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula. Substitute.], [#figure(figph[Two mathematical equations are displayed, illustrating the Pythagorean theorem. The first equation is a^2 + b^2 = c^2. The second equation shows specific values substituted: 5^2 + b^2 = 13^2.], alt: "Two mathematical equations are displayed, illustrating the Pythagorean theorem. The first equation is a^2 + b^2 = c^2. The second equation shows specific values substituted: 5^2 + b^2 = 13^2.", caption: none)], [Step 5. #strong[Solve] the equation. Isolate the variable term. Use the definition of the square root. #linebreak() Simplify.], [#figure(figph[Step-by-step solution for 'b' in the equation 25 + b^2 = 169. The calculation shows b^2 = 144 and concludes with b = 12, demonstrating fundamental algebraic problem-solving.], alt: "Step-by-step solution for 'b' in the equation 25 + b^2 = 169. The calculation shows b^2 = 144 and concludes with b = 12, demonstrating fundamental algebraic problem-solving.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[A step-by-step mathematical verification showing that 5 squared plus 12 squared equals 13 squared, confirming a Pythagorean triple.], alt: "A step-by-step mathematical verification showing that 5 squared plus 12 squared equals 13 squared, confirming a Pythagorean triple.", caption: none)], [], [Step 7. #strong[Answer] the question.], [The length of the leg is 12.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the Pythagorean Theorem to find the length of the leg. #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled “b”.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15, the other is labeled “b”.", caption: none) #solutionbox[ 8 ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Use the Pythagorean Theorem to find the length of the leg. #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled “b”.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 15. One of the sides touching the right angle is labeled as 9, the other is labeled “b”.", caption: none) #solutionbox[ 12 ] ] #examplebox("Example 9")[][ Kelvin is building a gazebo and wants to brace each corner by placing a #math.equation(block: false, alt: "10-inch")[$"10-inch"$] wooden bracket diagonally as shown. How far below the corner should he fasten the bracket if he wants the distances from the corner to each end of the bracket to be equal? Approximate to the nearest tenth of an inch. #figure(figph[A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.], alt: "A picture of a gazebo is shown. Beneath the roof is a rectangular shape. There are two braces from the top to each side. The brace on the left is labeled as 10 inches. From where the brace hits the side to the roof is labeled as x.", caption: none) #solutionbox[ #figure(table( columns: 2, align: left, inset: 6pt, table.header([Step 1. #strong[Read] the problem.], []), [Step 2. #strong[Identify] what you are looking for.], [the distance from the corner that the bracket should be attached], [Step 3. #strong[Name.] Choose a variable to represent it.], [Let #emph[x] = the distance from the corner #linebreak() #figure(figph[A right-angled isosceles triangle is depicted with two equal legs labeled 'x' and the hypotenuse measuring '10 inches'. This geometry problem asks for the value of x.], alt: "A right-angled isosceles triangle is depicted with two equal legs labeled 'x' and the hypotenuse measuring '10 inches'. This geometry problem asks for the value of x.", caption: none)], [Step 4. #strong[Translate.] #linebreak() Write the appropriate formula. #linebreak() Substitute.], [#linebreak() #figure(figph[Two mathematical equations are displayed: a squared plus b squared equals c squared, and x squared plus x squared equals 10 squared.], alt: "Two mathematical equations are displayed: a squared plus b squared equals c squared, and x squared plus x squared equals 10 squared.", caption: none)], [Step 5. #strong[Solve] the equation. #linebreak() Isolate the variable. #linebreak() Use the definition of the square root. #linebreak() Simplify. Approximate to the nearest tenth.], [#figure(figph[Mathematical calculation solving 2x^2 = 100, resulting in x = sqrt(50). The final step shows 'b' approximated as 7.1, indicating the numerical value of the square root of 50.], alt: "Mathematical calculation solving 2x^2 = 100, resulting in x = sqrt(50). The final step shows 'b' approximated as 7.1, indicating the numerical value of the square root of 50.", caption: none)], [Step 6. #strong[Check:] #linebreak() #figure(figph[The image displays two mathematical expressions related to the Pythagorean theorem. The first is the general formula a^2 + b^2 = c^2. Below it, a specific calculation is posed: (7.1)^2 + (7.1)^2 is approximately equal to 10^2, with a question mark indicating an inquiry.], alt: "The image displays two mathematical expressions related to the Pythagorean theorem. The first is the general formula a^2 + b^2 = c^2. Below it, a specific calculation is posed: (7.1)^2 + (7.1)^2 is approximately equal to 10^2, with a question mark indicating an inquiry.", caption: none) #linebreak() Yes.], [], [Step 7. #strong[Answer] the question.], [Kelvin should fasten each piece of wood approximately 7.1" from the corner.], )) ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ John puts the base of a #math.equation(block: false, alt: "13-ft")[$"13-ft"$] ladder #math.equation(block: false, alt: "5")[$5$] feet from the wall of his house. How far up the wall does the ladder reach? #figure(figph[A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.], alt: "A picture of a house is shown. There is a ladder leaning against the side of the house. The ladder is labeled 13 feet. The horizontal distance from the ladder's base to the house is labeled 5 feet.", caption: none) #solutionbox[ 12 feet ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ Randy wants to attach a #math.equation(block: false, alt: "17-ft")[$"17-ft"$] string of lights to the top of the #math.equation(block: false, alt: "15-ft")[$"15-ft"$] mast of his sailboat. How far from the base of the mast should he attach the end of the light string? #figure(figph[A picture of a boat is shown. The height of the center pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.], alt: "A picture of a boat is shown. The height of the center pole is labeled 15 feet. The string of lights is at a diagonal from the top of the pole and is labeled 17 feet.", caption: none) #solutionbox[ 8 feet ] ] #notebox("Note", rgb("#8a94a6"), rgb("#556666"), rgb("#f7f8fa"))[ #emph[ACCESS ADDITIONAL ONLINE RESOURCES] - #link("https://www.openstax.org/l/24sumintangles")[Animation: The Sum of the Interior Angles of a Triangle] - #link("https://www.openstax.org/l/24simpolygons")[Similar Polygons] - #link("https://www.openstax.org/l/24hyporighttri")[Example: Determine the Length of the Hypotenuse of a Right Triangle] ] === Key Concepts - #emph[Supplementary and Complementary Angles]- If the sum of the measures of two angles is 180°, then the angles are supplementary. - If #math.equation(block: false, alt: "∠ A")[$"∠" A$] and #math.equation(block: false, alt: "∠ B")[$"∠" B$] are supplementary, then #math.equation(block: false, alt: "m ∠ A plus m ∠ B equals 180")[$m "∠" A + m "∠" B = 180$]. - If the sum of the measures of two angles is 90°, then the angles are complementary. - If #math.equation(block: false, alt: "∠ A")[$"∠" A$] and #math.equation(block: false, alt: "∠ B")[$"∠" B$] are complementary, then #math.equation(block: false, alt: "m ∠ A plus m ∠ B equals 90")[$m "∠" A + m "∠" B = 90$]. - #emph[Solve Geometry Applications]+ Read the problem and make sure you understand all the words and ideas. Draw a figure and label it with the given information. + Identify what you are looking for. + Name what you are looking for and choose a variable to represent it. + Translate into an equation by writing the appropriate formula or model for the situation. Substitute in the given information. + Solve the equation using good algebra techniques. + Check the answer in the problem and make sure it makes sense. + Answer the question with a complete sentence. - #emph[Sum of the Measures of the Angles of a Triangle]#figure(figph[The image shows a triangle with its points labeled A, B, and C.], alt: "The image shows a triangle with its points labeled A, B, and C.", caption: none) - For any #math.equation(block: false, alt: "Δ A B C ,")[$"Δ" A B C ,$] the sum of the measures is 180° - #math.equation(block: false, alt: "m ∠ A plus m ∠ B plus m ∠ C equals 180")[$m "∠" A + m "∠" B + m "∠" C = 180$] - #emph[Right Triangle] #figure(figph[A black and white diagram of a right-angled triangle, showing the 90-degree angle at one vertex marked with a square symbol and the label '90°' inside it.], alt: "A black and white diagram of a right-angled triangle, showing the 90-degree angle at one vertex marked with a square symbol and the label '90°' inside it.", caption: none) - A right triangle is a triangle that has one 90° angle, which is often marked with a #math.equation(block: false, alt: "⦜")[$"⦜"$]symbol. - #emph[Properties of Similar Triangles]- If two triangles are similar, then their corresponding angle measures are equal and their corresponding side lengths have the same ratio. ==== Practice Makes Perfect #strong[Use the Properties of Angles] In the following exercises, find ⓐ the supplement and ⓑ the complement of the given angle. #math.equation(block: true, alt: "53°")[$"53°"$] #solutionbox[ + ⓐ 127° + ⓑ 37° ] #math.equation(block: true, alt: "16°")[$"16°"$] #math.equation(block: true, alt: "29°")[$"29°"$] #solutionbox[ + ⓐ 151° + ⓑ 61° ] #math.equation(block: true, alt: "72°")[$"72°"$] In the following exercises, use the properties of angles to solve. Find the supplement of a #math.equation(block: false, alt: "135°")[$"135°"$] angle. #solutionbox[ 45° ] Find the complement of a #math.equation(block: false, alt: "38°")[$"38°"$] angle. Find the complement of a #math.equation(block: false, alt: "27.5°")[$27.5°$] angle. #solutionbox[ 62.5° ] Find the supplement of a #math.equation(block: false, alt: "109.5°")[$109.5°$] angle. Two angles are supplementary. The larger angle is #math.equation(block: false, alt: "56°")[$"56°"$] more than the smaller angle. Find the measures of both angles. #solutionbox[ 62°, 118° ] Two angles are supplementary. The smaller angle is #math.equation(block: false, alt: "36°")[$"36°"$] less than the larger angle. Find the measures of both angles. Two angles are complementary. The smaller angle is #math.equation(block: false, alt: "34°")[$"34°"$] less than the larger angle. Find the measures of both angles. #solutionbox[ 62°, 28° ] Two angles are complementary. The larger angle is #math.equation(block: false, alt: "52°")[$"52°"$] more than the smaller angle. Find the measures of both angles. #strong[Use the Properties of Triangles] In the following exercises, solve using properties of triangles. The measures of two angles of a triangle are #math.equation(block: false, alt: "26°")[$"26°"$] and #math.equation(block: false, alt: "98° .")[$"98°" .$] Find the measure of the third angle. #solutionbox[ 56° ] The measures of two angles of a triangle are #math.equation(block: false, alt: "61°")[$"61°"$] and #math.equation(block: false, alt: "84° .")[$"84°" .$] Find the measure of the third angle. The measures of two angles of a triangle are #math.equation(block: false, alt: "105°")[$"105°"$] and #math.equation(block: false, alt: "31° .")[$"31°" .$] Find the measure of the third angle. #solutionbox[ 44° ] The measures of two angles of a triangle are #math.equation(block: false, alt: "47°")[$"47°"$] and #math.equation(block: false, alt: "72° .")[$"72°" .$] Find the measure of the third angle. One angle of a right triangle measures #math.equation(block: false, alt: "33° .")[$"33°" .$] What is the measure of the other angle? #solutionbox[ 57° ] One angle of a right triangle measures #math.equation(block: false, alt: "51° .")[$"51°" .$] What is the measure of the other angle? One angle of a right triangle measures #math.equation(block: false, alt: "22.5 ° .")[$22.5 ° .$] What is the measure of the other angle? #solutionbox[ 67.5° ] One angle of a right triangle measures #math.equation(block: false, alt: "36.5 ° .")[$36.5 ° .$] What is the measure of the other angle? The two smaller angles of a right triangle have equal measures. Find the measures of all three angles. #solutionbox[ 45°, 45°, 90° ] The measure of the smallest angle of a right triangle is #math.equation(block: false, alt: "20°")[$"20°"$] less than the measure of the other small angle. Find the measures of all three angles. The angles in a triangle are such that the measure of one angle is twice the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles. #solutionbox[ 30°, 60°, 90° ] The angles in a triangle are such that the measure of one angle is #math.equation(block: false, alt: "20°")[$"20°"$] more than the measure of the smallest angle, while the measure of the third angle is three times the measure of the smallest angle. Find the measures of all three angles. #strong[Find the Length of the Missing Side] In the following exercises, #math.equation(block: false, alt: "Δ A B C")[$"Δ" A B C$] is similar to #math.equation(block: false, alt: "Δ X Y Z .")[$"Δ" X Y Z .$] Find the length of the indicated side. #figure(figph[Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10.], alt: "Two triangles are shown. They appear to be the same shape, but the triangle on the right is smaller. The vertices of the triangle on the left are labeled A, B, and C. The side across from A is labeled 9, the side across from B is labeled b, and the side across from C is labeled 15. The vertices of the triangle on the right are labeled X, Y, and Z. The side across from X is labeled x, the side across from Y is labeled 8, and the side across from Z is labeled 10.", caption: none) side #math.equation(block: false, alt: "b")[$b$] #solutionbox[ 12 ] side #math.equation(block: false, alt: "x")[$x$] On a map, San Francisco, Las Vegas, and Los Angeles form a triangle whose sides are shown in the figure below. The actual distance from Los Angeles to Las Vegas is #math.equation(block: false, alt: "270")[$270$] miles. #linebreak() #figure(figph[A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.], alt: "A triangle is shown. The vertices are labeled San Francisco, Las Vegas, and Los Angeles. The side across from San Francisco is labeled 1 inch, the side across from Las Vegas is labeled 1.3 inches, and the side across from Los Angeles is labeled 2.1 inches.", caption: none) Find the distance from Los Angeles to San Francisco. #solutionbox[ 351 miles ] Find the distance from San Francisco to Las Vegas. #strong[Use the Pythagorean Theorem] In the following exercises, use the Pythagorean Theorem to find the length of the hypotenuse. #figure(figph[A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.], alt: "A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 9, the other as 12.", caption: none) #solutionbox[ 15 ] #figure(figph[A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.], alt: "A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 16, the other as 12.", caption: none) #figure(figph[A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.], alt: "A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 15, the other as 20.", caption: none) #solutionbox[ 25 ] #figure(figph[A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.], alt: "A right triangle is shown. The right angle is marked with a box. One of the sides touching the right angle is labeled as 5, the other as 12.", caption: none) #strong[Find the Length of the Missing Side] In the following exercises, use the Pythagorean Theorem to find the length of the missing side. Round to the nearest tenth, if necessary. #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 10. One of the sides touching the right angle is labeled as 6.", caption: none) #solutionbox[ 8 ] #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 8.", caption: none) #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 5.", caption: none) #solutionbox[ 12 ] #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 20. One of the sides touching the right angle is labeled as 16.", caption: none) #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 13. One of the sides touching the right angle is labeled as 8.", caption: none) #solutionbox[ 10.2 ] #figure(figph[A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.], alt: "A right triangle is shown. The right angle is marked with a box. Both of the sides touching the right angle are labeled as 6.", caption: none) #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 17. One of the sides touching the right angle is labeled as 15.", caption: none) #solutionbox[ 16.2 ] #figure(figph[A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.], alt: "A right triangle is shown. The right angle is marked with a box. The side across from the right angle is labeled as 7. One of the sides touching the right angle is labeled as 5.", caption: none) In the following exercises, solve. Approximate to the nearest tenth, if necessary. A #math.equation(block: false, alt: "13-foot")[$"13-foot"$] string of lights will be attached to the top of a #math.equation(block: false, alt: "12-foot")[$"12-foot"$] pole for a holiday display. How far from the base of the pole should the end of the string of lights be anchored? #figure(figph[A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.], alt: "A vertical pole is shown with a string of lights going from the top of the pole to the ground. The pole is labeled 12 feet. The string of lights is labeled 13 feet.", caption: none) #solutionbox[ 5 feet ] Pam wants to put a banner across her garage door to congratulate her son on his college graduation. The garage door is #math.equation(block: false, alt: "12")[$12$] feet high and #math.equation(block: false, alt: "16")[$16$] feet wide. How long should the banner be to fit the garage door? #figure(figph[A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.], alt: "A picture of a house is shown. The rectangular garage is 12 feet high and 16 feet wide. A blue banner goes diagonally across the garage.", caption: none) Chi is planning to put a path of paving stones through her flower garden. The flower garden is a square with sides of #math.equation(block: false, alt: "10")[$10$] feet. What will the length of the path be? #figure(figph[A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.], alt: "A square garden is shown. One side is labeled as 10 feet. There is a diagonal path of blue circular stones going from the lower left corner to the upper right corner.", caption: none) #solutionbox[ 14.1 feet ] Brian borrowed a #math.equation(block: false, alt: "20-foot")[$"20-foot"$] extension ladder to paint his house. If he sets the base of the ladder #math.equation(block: false, alt: "6")[$6$] feet from the house, how far up will the top of the ladder reach? #figure(figph[A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.], alt: "A picture of a house is shown with a ladder leaning against it. The ladder is labeled 20 feet tall. The horizontal distance from the house to the base of the ladder is 6 feet.", caption: none) ==== Everyday Math #strong[Building a scale model] Joe wants to build a doll house for his daughter. He wants the doll house to look just like his house. His house is #math.equation(block: false, alt: "30")[$30$] feet wide and #math.equation(block: false, alt: "35")[$35$] feet tall at the highest point of the roof. If the dollhouse will be #math.equation(block: false, alt: "2.5")[$2.5$] feet wide, how tall will its highest point be? #solutionbox[ 2.9 feet ] #strong[Measurement] A city engineer plans to build a footbridge across a lake from point #math.equation(block: false, alt: "X")[$"X"$] to point #math.equation(block: false, alt: "Y ,")[$"Y" ,$] as shown in the picture below. To find the length of the footbridge, she draws a right triangle #math.equation(block: false, alt: "XYZ ,")[$"XYZ" ,$] with right angle at #math.equation(block: false, alt: "X .")[$"X" .$] She measures the distance from #math.equation(block: false, alt: "X")[$"X"$] to #math.equation(block: false, alt: "Z , 800")[$"Z" , 800$] feet, and from #math.equation(block: false, alt: "Y")[$"Y"$] to #math.equation(block: false, alt: "Z , 1,000")[$"Z" , 1,000$] feet. How long will the bridge be? #figure(figph[A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.], alt: "A lake is shown. Point Y is on one side of the lake, directly across from point X. Point Z is on the same side of the lake as point X.", caption: none) ==== Writing Exercises Write three of the properties of triangles from this section and then explain each in your own words. #solutionbox[ Answers will vary. ] Explain how the figure below illustrates the Pythagorean Theorem for a triangle with legs of length #math.equation(block: false, alt: "3")[$3$] and #math.equation(block: false, alt: "4 .")[$4 .$] #figure(figph[Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.], alt: "Three squares are shown, forming a right triangle in the center. Each square is divided into smaller squares. The smallest square is divided into 9 small squares. The medium square is divided into 16 small squares. The large square is divided into 25 small squares.", caption: none) ==== Self Check ⓐ After completing the exercises, use this checklist to evaluate your mastery of the objectives of this section. #figure(figph[A self-assessment table for geometry skills, listing 'I can...' statements for properties of angles, triangles, and the Pythagorean Theorem, with options to rate understanding as 'Confidentially', 'With some help', or 'No-I don't get it!'.], alt: "A self-assessment table for geometry skills, listing 'I can...' statements for properties of angles, triangles, and the Pythagorean Theorem, with options to rate understanding as 'Confidentially', 'With some help', or 'No-I don't get it!'.", caption: none) ⓑ What does this checklist tell you about your mastery of this section? What steps will you take to improve?