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== 10.3#h(0.6em)Triangles


  #figure(figph[A view of an arched ceiling in an architectural building.], alt: "A view of an arched ceiling in an architectural building.", caption: [The appearance of triangles in buildings is part of modern-day architectural design.])


  
    
=== Learning Objectives


    After completing this section, you should be able to:


    + Identify triangles by their sides.
+ Identify triangles by their angles.
+ Determine if triangles are congruent.
+ Determine if triangles are similar.
+ Find the missing side of similar triangles.


  
  How were the ancient Greeks able to calculate the radius of Earth? How did soldiers gauge their target? How was it possible centuries ago to estimate the height of a sail at sea? Triangles have always played a significant role in how we find heights of objects too high to measure or distances between objects too far away to calculate. In particular, the concept of similar triangles has countless applications in the real world, and we shall explore some of those applications in this section.


  Technology has given us instruments that allow us to find measurements of distant objects with little effort. However, it is all based on the properties of triangles discovered centuries ago. In this section, we will explore the various types of triangles and their special properties, as well as how to measure interior and exterior angles. We will also explore congruence theorems and similarity.


  
    
=== Identifying Triangles


    Joining any three noncollinear points with line segments produces a triangle. For example, given points #math.equation(block: false, alt: "A")[$A$], #math.equation(block: false, alt: "B")[$B$], and #math.equation(block: false, alt: "C")[$C$], connected by the line segments #math.equation(block: false, alt: "A B ¯ , B C , ¯")[$overline(A B) , overline(B C ",")$] and #math.equation(block: false, alt: "A C ¯ ,")[$overline(A C) ,$] we have a triangle, as shown.


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    Triangles are classified by their angles and their sides. All angles in an acute triangle measure #math.equation(block: false, alt: "less than 90 degrees .")[$< 90^(∘) .$] One of the angles in a right triangle measures #math.equation(block: false, alt: "90 degrees ,")[$90^(∘) ,$] symbolized by □. One angle in an obtuse triangle measures between #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] and #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$] Sides that have equal length are indicated by the same hash marks.  illustrates the shapes of the basic triangles, their names, and their properties.


    A few other facts to remember as we move forward:


    - The points where the line segments meet are called the #strong[vertices] (plural for #strong[vertex]).
- We often refer to sides of a triangle by the angle they are opposite. In other words, side #math.equation(block: false, alt: "a")[$a$] is opposite angle #math.equation(block: false, alt: "A")[$A$], side #math.equation(block: false, alt: "b")[$b$] is opposite angle #math.equation(block: false, alt: "B")[$B$], and side #math.equation(block: false, alt: "c")[$c$] is opposite angle #math.equation(block: false, alt: "C")[$C$].


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)


    We want to add a special note about right triangles here, as they are referred to more than any other triangle. The side opposite the right angle is its longest side and is called the #strong[hypotenuse], and the sides adjacent to the right angle are called the legs.


    One of the most important properties of triangles is that the sum of the interior angles equals #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$] Euclid discovered and proved this property using parallel lines. The completed sketch is shown.


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)


    This is how the proof goes:


    #strong[Step 1:] Start with a straight line #math.equation(block: false, alt: "A B ↔")[$accent(A B, ↔)$] and a point #math.equation(block: false, alt: "C")[$C$] not on the line.


    #strong[Step 2:] Draw a line through point #math.equation(block: false, alt: "C")[$C$] parallel to the line #math.equation(block: false, alt: "A B ↔")[$accent(A B, ↔)$].


    #strong[Step 3:] Construct two transversals (a line crossing the parallel lines), one angled to the right and one angled to the left, to intersect the parallel lines.


    #strong[Step 4:] Because of the property that alternate interior angles inside parallel lines are equal, we have that


    #math.equation(block: true, alt: "m ∡ 2 equals m ∢ 1 and m ∡ 3 equals m ∡ 4.")[$m ∡ 2 = m ∢ 1 #h(1em) "and" #h(1em) m ∡ 3 = m ∡ 4.$]
    #strong[Step 5:] Notice that #math.equation(block: false, alt: "m ∡ 2 plus m ∡ 5 plus m ∡ 4 equals 180 degrees")[$m ∡ 2 + m ∡ 5 + m ∡ 4 = 180^(∘)$] by the straight angle property.


    #strong[Step 6:] Therefore, by substitution, #math.equation(block: false, alt: "m ∡ 1 equals m ∡ 2 ,")[$m ∡ 1 = m ∡ 2 ,$] and #math.equation(block: false, alt: "m ∡ 3 equals m ∡ 4 ,")[$m ∡ 3 = m ∡ 4 ,$] we have that


    #math.equation(block: true, alt: "m ∡ 1 plus m ∡ 3 plus m ∢ 5 equals 180 degrees .")[$m ∡ 1 + m ∡ 3 + m ∢ 5 = 180^(∘) "."$]
    Therefore, the sum of the interior angles of a #math.equation(block: false, alt: "triangle equals 180 degrees .")[$"triangle" = 180^(∘) .$]


    #examplebox("Example 1")[Finding Measures of Angles Inside a Triangle][
Find the measure of each angle in the triangle shown . We know that the sum of the angles must equal #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$]


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      #solutionbox[
#strong[Step 1:] As the sum of the interior angles equals #math.equation(block: false, alt: "180 degrees ,")[$180^(∘) ,$] we can use algebra to find the measures:


        #math.equation(block: true, alt: "x plus open parenthesis x plus 17 close parenthesis plus open parenthesis 3 x minus 62 close parenthesis, equals, 180; 5 x minus 45, equals, 180; 5 x, equals, 225; x, equals, the fraction 225 over 5 equals 45")[$x + ( x + 17 ) + ( 3 x − 62 ) & = & 180 \ 5 x − 45 & = & 180 \ 5 x & = & 225 \ x & = & frac(225, 5) = 45$]
        #strong[Step 2:] Now that we have the value of #math.equation(block: false, alt: "x")[$x$], we can substitute 45 into the other two expressions to find the measure of those angles:


        #math.equation(block: true, alt: "open parenthesis x plus 17 close parenthesis, equals, 45 plus 17 equals 62 degrees; open parenthesis 3 x minus 62 close parenthesis, equals, 3 open parenthesis 45 close parenthesis minus 62 equals 135 minus 62 equals 73 degrees")[$( x + 17 ) & = & 45 + 17 = 62^(∘) \ ( 3 x − 62 ) & = & 3 ( 45 ) − 62 = 135 − 62 = 73^(∘)$]
        #strong[Step 3:] Then, #math.equation(block: false, alt: "m ∡ x equals 45 degrees")[$m ∡ x = 45^(∘)$], #math.equation(block: false, alt: "m ∡ open parenthesis x plus 17 close parenthesis equals 62 degrees")[$m ∡ ( x + 17 ) = 62^(∘)$], and #math.equation(block: false, alt: "m ∡ open parenthesis 3 x minus 62 close parenthesis equals 73 degrees .")[$m ∡ ( 3 x − 62 ) = 73^(∘) .$]
]
]


    
    #examplebox("Example 2")[Finding Angle Measures][
Find the measure of angles numbered 1–5 in .


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)


      
      #solutionbox[
The #math.equation(block: false, alt: "m ∡ 1 equals 119 degrees")[$m ∡ 1 = 119^(∘)$] because it is supplementary with the unknown angle of the adjacent triangle. The unknown angle measures #math.equation(block: false, alt: "61 degrees .")[$61^(∘) .$] The #math.equation(block: false, alt: "m ∡ 2 equals 61 degrees")[$m ∡ 2 = 61^(∘)$] because of vertical angles. The #math.equation(block: false, alt: "m ∡ 5 equals 59 degrees")[$m ∡ 5 = 59^(∘)$] because the angle that is supplementary to the #math.equation(block: false, alt: "134 degrees")[$134^(∘)$] measures #math.equation(block: false, alt: "46 degrees")[$46^(∘)$], and angle 5 is the unknown angle in that triangle. The #math.equation(block: false, alt: "m ∡ 4 equals 59 degrees")[$m ∡ 4 = 59^(∘)$] by vertical angles. Finally, #math.equation(block: false, alt: "m ∡ 3 equals 60 degrees ,")[$m ∡ 3 = 60^(∘) ,$] as it is the third angle in the triangle with angles measuring #math.equation(block: false, alt: "59 degrees")[$59^(∘)$] and #math.equation(block: false, alt: "61 degrees .")[$61^(∘) .$]
]
]


    
  
  
    
=== Congruence


    If two triangles have equal angles and their sides lengths are equal, the triangles are congruent. In other words, if you can pick up one triangle and place it on top of the other triangle and they coincide, even if you have to rotate one, they are congruent.


    #examplebox("Example 3")[Determining If Triangles Are Congruent][
In , is the triangle #math.equation(block: false, alt: "ABC")[$italic("ABC")$] congruent to triangle #math.equation(block: false, alt: "DEF")[$italic("DEF")$]?


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  stdfig([#grid(columns: 2, gutter: 1.5em,
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 180.67pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle ABC])
  line((33.33pt, 33.33pt), (166.67pt, 33.33pt), (100.00pt, 133.33pt), close: true, fill: none, stroke: 0.7pt + black)
  line((130.01pt, 81.11pt), (136.66pt, 85.55pt), stroke: 0.7pt + black)
  line((69.99pt, 81.11pt), (63.34pt, 85.55pt), stroke: 0.7pt + black)
  arc((33.33pt, 33.33pt), start: 0.00deg, stop: 56.31deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((166.67pt, 33.33pt), start: 123.69deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((37.33pt, 37.33pt), anchor: "south-west", text(size: 9pt)[A])
  content((162.67pt, 37.33pt), anchor: "south-east", text(size: 9pt)[C])
  content((100.00pt, 129.33pt), anchor: "north", text(size: 9pt)[B])
})],
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 180.67pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle DEF])
  line((33.33pt, 33.33pt), (166.67pt, 33.33pt), (100.00pt, 133.33pt), close: true, fill: none, stroke: 0.7pt + black)
  line((130.01pt, 81.11pt), (136.66pt, 85.55pt), stroke: 0.7pt + black)
  line((69.99pt, 81.11pt), (63.34pt, 85.55pt), stroke: 0.7pt + black)
  arc((33.33pt, 33.33pt), start: 0.00deg, stop: 56.31deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((166.67pt, 33.33pt), start: 123.69deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((37.33pt, 37.33pt), anchor: "south-west", text(size: 9pt)[D])
  content((162.67pt, 37.33pt), anchor: "south-east", text(size: 9pt)[F])
  content((100.00pt, 129.33pt), anchor: "north", text(size: 9pt)[E])
})]
)]),
  alt: "Two triangles, A B C and D E F. Angles A and C are equal and congruent to angles D and F. The sides, A B and B C are equal. The side, A B is congruent to the side, D E. The sides, D E and E F are equal. The side, B C is congruent to the side, E F.", caption: none,
)


      
      #solutionbox[
Triangle #math.equation(block: false, alt: "ABC")[$italic("ABC")$] is congruent to triangle #math.equation(block: false, alt: "DEF")[$italic("DEF")$]. Angles #math.equation(block: false, alt: "A")[$A$] and #math.equation(block: false, alt: "C")[$C$] are congruent to angles #math.equation(block: false, alt: "D")[$D$] and #math.equation(block: false, alt: "F")[$F$], which implies that angle #math.equation(block: false, alt: "B")[$B$] is congruent to angle #math.equation(block: false, alt: "E")[$E$]. Side #math.equation(block: false, alt: "AB")[$italic("AB")$] is congruent to side #math.equation(block: false, alt: "DE")[$italic("DE")$], and side #math.equation(block: false, alt: "CB")[$italic("CB")$] is congruent to side #math.equation(block: false, alt: "FE")[$italic("FE")$], which implies that side #math.equation(block: false, alt: "AC")[$italic("AC")$] is congruent to side #math.equation(block: false, alt: "DF")[$italic("DF")$].
]
]


    
  
  
    
=== The Congruence Theorems


    The following theorems are tools you can use to prove that two triangles are congruent. We use the symbol #math.equation(block: false, alt: "≅")[$≅$] to define congruence. For example, #math.equation(block: false, alt: "Δ A B C ≅ Δ D E F")[$"Δ" A B C ≅ "Δ" D E F$].


    #strong[Side-Side-Side (SSS).] If three sides of one triangle are equal to the corresponding sides of the second triangle, then the triangles are congruent.


    #figure(
  stdfig([#grid(columns: 2, gutter: 1.5em,
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 180.67pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle DEF])
  line((33.33pt, 33.33pt), (166.67pt, 33.33pt), (100.00pt, 133.33pt), close: true, fill: none, stroke: 0.7pt + black)
  line((100.00pt, 37.33pt), (100.00pt, 29.33pt), stroke: 0.7pt + black)
  line((130.70pt, 80.07pt), (137.35pt, 84.51pt), stroke: 0.7pt + black)
  line((129.31pt, 82.15pt), (135.97pt, 86.59pt), stroke: 0.7pt + black)
  line((71.38pt, 83.19pt), (64.73pt, 87.63pt), stroke: 0.7pt + black)
  line((69.99pt, 81.11pt), (63.34pt, 85.55pt), stroke: 0.7pt + black)
  line((68.61pt, 79.03pt), (61.95pt, 83.47pt), stroke: 0.7pt + black)
  content((29.33pt, 29.33pt), anchor: "north-east", text(size: 9pt)[D])
  content((170.67pt, 29.33pt), anchor: "north-west", text(size: 9pt)[E])
  content((100.00pt, 137.33pt), anchor: "south", text(size: 9pt)[F])
})],
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 180.67pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle RST])
  line((33.33pt, 33.33pt), (166.67pt, 33.33pt), (100.00pt, 133.33pt), close: true, fill: none, stroke: 0.7pt + black)
  line((100.00pt, 37.33pt), (100.00pt, 29.33pt), stroke: 0.7pt + black)
  line((130.70pt, 80.07pt), (137.35pt, 84.51pt), stroke: 0.7pt + black)
  line((129.31pt, 82.15pt), (135.97pt, 86.59pt), stroke: 0.7pt + black)
  line((71.38pt, 83.19pt), (64.73pt, 87.63pt), stroke: 0.7pt + black)
  line((69.99pt, 81.11pt), (63.34pt, 85.55pt), stroke: 0.7pt + black)
  line((68.61pt, 79.03pt), (61.95pt, 83.47pt), stroke: 0.7pt + black)
  content((29.33pt, 29.33pt), anchor: "north-east", text(size: 9pt)[R])
  content((170.67pt, 29.33pt), anchor: "north-west", text(size: 9pt)[S])
  content((100.00pt, 137.33pt), anchor: "south", text(size: 9pt)[T])
})]
)]),
  alt: "Two triangles, D E F and R S T. The side, D F is congruent to the side, R T. The side, E F is congruent to the side, S T. The side, D E is congruent to the side, R S.", caption: [Side-Side-Side (SSS)],
)


    We have that #math.equation(block: false, alt: "D F ¯ ≅ R T ¯")[$overline(D F) ≅ overline(R T)$], #math.equation(block: false, alt: "E F ¯ ≅ S T ¯ ,")[$overline(E F) ≅ overline(S T) ,$] and #math.equation(block: false, alt: "DE ¯ ≅ RS ¯ ,")[$overline(italic("DE")) ≅ overline(italic("RS")) ","$] then #math.equation(block: false, alt: "Δ D E F ≅ Δ R S T .")[$"Δ" D E F ≅ "Δ" R S T .$]


    #strong[Side-Angle-Side (SAS).] If two sides of a triangle and the angle between them are equal to the corresponding two sides and included angle of the second triangle, then the triangles are congruent. We see that #math.equation(block: false, alt: "A B ¯ ≅ A prime B prime ¯")[$overline(A B) ≅ overline(A^(′) B^(′))$] and #math.equation(block: false, alt: "B C ¯ ≅ B prime C prime ¯")[$overline(B C) ≅ overline(B^(′) C^(′))$], #math.equation(block: false, alt: "m ∡ B equals m ∡ B prime")[$m ∡ B = m ∡ B^(′)$], then #math.equation(block: false, alt: "Δ A B C ≅ Δ A prime B prime C prime")[$"Δ" A B C ≅ "Δ" A^(′) B^(′) C^(′)$].


    #figure(
  stdfig([#grid(columns: 2, gutter: 1.5em,
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 180.67pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle ABC])
  line((33.33pt, 33.33pt), (166.67pt, 33.33pt), (66.67pt, 133.33pt), close: true, fill: none, stroke: 0.7pt + black)
  line((98.75pt, 37.33pt), (98.75pt, 29.33pt), stroke: 0.7pt + black)
  line((101.25pt, 37.33pt), (101.25pt, 29.33pt), stroke: 0.7pt + black)
  line((53.79pt, 82.07pt), (46.21pt, 84.60pt), stroke: 0.7pt + black)
  arc((33.33pt, 33.33pt), start: 0.00deg, stop: 71.57deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((66.67pt, 137.33pt), anchor: "south", text(size: 9pt)[A])
  content((29.33pt, 29.33pt), anchor: "north-east", text(size: 9pt)[B])
  content((170.67pt, 29.33pt), anchor: "north-west", text(size: 9pt)[C])
})],
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 180.67pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle A′B′C′])
  line((33.33pt, 33.33pt), (166.67pt, 33.33pt), (66.67pt, 133.33pt), close: true, fill: none, stroke: 0.7pt + black)
  line((98.75pt, 37.33pt), (98.75pt, 29.33pt), stroke: 0.7pt + black)
  line((101.25pt, 37.33pt), (101.25pt, 29.33pt), stroke: 0.7pt + black)
  line((53.79pt, 82.07pt), (46.21pt, 84.60pt), stroke: 0.7pt + black)
  arc((33.33pt, 33.33pt), start: 0.00deg, stop: 71.57deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((66.67pt, 137.33pt), anchor: "south", text(size: 9pt)[A′])
  content((29.33pt, 29.33pt), anchor: "north-east", text(size: 9pt)[B′])
  content((170.67pt, 29.33pt), anchor: "north-west", text(size: 9pt)[C′])
})]
)]),
  alt: "Two triangles, A B C and A prime B prime C prime. The angles, B and B prime are congruent. The side, A B is congruent to the side, A prime B prime. The side, B C is congruent to the side, B prime C prime.", caption: [Side-Angle-Side (SAS)],
)


    #strong[Angle-Side-Angle (ASA).] If two angles and the side between them in one triangle are congruent to the two corresponding angles and the side between them in a second triangle, then the two triangles are congruent. Notice that #math.equation(block: false, alt: "m ∡ A ≅ m ∡ F")[$m ∡ "A" ≅ m ∡ "F"$], and #math.equation(block: false, alt: "m ∡ C ≅ m ∡ D")[$m ∡ "C" ≅ m ∡ "D"$], #math.equation(block: false, alt: "AC ¯ ≅ DF ¯")[$overline(italic("AC")) ≅ overline(italic("DF"))$], then #math.equation(block: false, alt: "Δ ABC ≅ Δ DEF .")[$"Δ" italic("ABC") ≅ "Δ" italic("DEF") .$]


    #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 104.91pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[Triangles ABC and DEF])
  line((18.18pt, 18.18pt), (90.91pt, 18.18pt), (36.36pt, 72.73pt), close: true, fill: none, stroke: 0.7pt + black)
  line((54.55pt, 22.18pt), (54.55pt, 14.18pt), stroke: 0.7pt + black)
  line((109.09pt, 18.18pt), (181.82pt, 18.18pt), (127.27pt, 72.73pt), close: true, fill: none, stroke: 0.7pt + black)
  line((145.45pt, 22.18pt), (145.45pt, 14.18pt), stroke: 0.7pt + black)
  arc((90.91pt, 18.18pt), start: 135.00deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((181.82pt, 18.18pt), start: 135.00deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((18.18pt, 18.18pt), start: 0.00deg, stop: 71.57deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((18.18pt, 18.18pt), start: 0.00deg, stop: 71.57deg, radius: 17.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((109.09pt, 18.18pt), start: 0.00deg, stop: 71.57deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((109.09pt, 18.18pt), start: 0.00deg, stop: 71.57deg, radius: 17.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((22.18pt, 22.18pt), anchor: "south-west", text(size: 9pt)[C])
  content((86.91pt, 22.18pt), anchor: "south-east", text(size: 9pt)[A])
  content((36.36pt, 68.73pt), anchor: "north", text(size: 9pt)[B])
  content((113.09pt, 22.18pt), anchor: "south-west", text(size: 9pt)[D])
  content((177.82pt, 22.18pt), anchor: "south-east", text(size: 9pt)[F])
  content((127.27pt, 68.73pt), anchor: "north", text(size: 9pt)[E])
})]),
  alt: "Two triangles, A B C and D E F. The sides, C A and D F rest on the same line. The sides, C A and D F are equal. The angles, A and F are congruent. The angles, C and D are congruent.", caption: [Angle-Side-Angle (ASA)],
)


    #strong[Angle-Angle-Side (AAS).] If two angles and a nonincluded side of one triangle are congruent to two angles and the nonincluded corresponding side of a second triangle, then the triangles are congruent. We see that #math.equation(block: false, alt: "m ∡ X ≅ m X prime")[$m ∡ X ≅ m X^(′)$], #math.equation(block: false, alt: "m ∡ Z ≅ m ∡ Z prime")[$m ∡ Z ≅ m ∡ Z^(′)$], and #math.equation(block: false, alt: "X Y ¯ ≅ X prime Y prime ¯")[$overline(X Y) ≅ overline(X^(′) Y^(′))$], then #math.equation(block: false, alt: "Δ X Y Z ≅ Δ X prime Y prime Z prime")[$"Δ" X Y Z ≅ "Δ" X^(′) Y^(′) Z^(′)$].


    #figure(
  stdfig([#grid(columns: 2, gutter: 1.5em,
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 128.29pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle XYZ])
  line((28.57pt, 28.57pt), (171.43pt, 28.57pt), (125.71pt, 68.57pt), close: true, fill: none, stroke: 0.7pt + black)
  content((95.13pt, 21.00pt), anchor: "north", text(size: 9pt)[5 cm])
  arc((28.57pt, 28.57pt), start: 0.00deg, stop: 22.38deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((51.13pt, 33.03pt), text(size: 9pt)[22°])
  arc((125.71pt, 68.57pt), start: 202.38deg, stop: 318.81deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((121.96pt, 45.88pt), text(size: 9pt)[118°])
  content((32.57pt, 32.57pt), anchor: "south-west", text(size: 9pt)[X])
  content((167.43pt, 32.57pt), anchor: "south-east", text(size: 9pt)[Y])
  content((125.71pt, 64.57pt), anchor: "north", text(size: 9pt)[Z])
})],
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 128.29pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle X′Y′Z′])
  line((28.57pt, 28.57pt), (171.43pt, 28.57pt), (125.71pt, 68.57pt), close: true, fill: none, stroke: 0.7pt + black)
  content((95.13pt, 21.00pt), anchor: "north", text(size: 9pt)[5 cm])
  arc((28.57pt, 28.57pt), start: 0.00deg, stop: 22.38deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((51.13pt, 33.03pt), text(size: 9pt)[22°])
  arc((125.71pt, 68.57pt), start: 202.38deg, stop: 318.81deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((121.96pt, 45.88pt), text(size: 9pt)[118°])
  content((32.57pt, 32.57pt), anchor: "south-west", text(size: 9pt)[X′])
  content((167.43pt, 32.57pt), anchor: "south-east", text(size: 9pt)[Y′])
  content((125.71pt, 64.57pt), anchor: "north", text(size: 9pt)[Z′])
})]
)]),
  alt: "Two triangles, X Y Z and X prime Y prime Z prime. In the triangle X Y Z, the side X Y measures 5 centimeters, and the angles X and Z measure 22 degrees and 118 degrees. In the triangle, X prime Y prime Z prime, the side X prime Y prime measures 5 centimeters, and the angles X prime ad Z prime measure 22 degrees and 118 degrees.", caption: [Angle-Angle-Side (AAS)],
)


    #examplebox("Example 4")[Identifying Congruence Theorems][
What congruence theorem is illustrated in ?


        #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 164.00pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[Rectangle formed by two right triangles])
  line((25.00pt, 25.00pt), (175.00pt, 25.00pt), (175.00pt, 125.00pt), (25.00pt, 125.00pt), close: true, fill: none, stroke: 0.7pt + black)
  line((25.00pt, 125.00pt), (175.00pt, 25.00pt), stroke: 0.7pt + black)
  line((25.00pt, 125.00pt), (25.00pt, 117.00pt), (33.00pt, 117.00pt), (33.00pt, 125.00pt), close: true, stroke: 0.6pt + black, fill: none)
  line((175.00pt, 25.00pt), (175.00pt, 33.00pt), (167.00pt, 33.00pt), (167.00pt, 25.00pt), close: true, stroke: 0.6pt + black, fill: none)
})]),
  alt: "A rectangle is formed by joining two right triangles. The hypotenuses of both the triangles share the same side. The top-left and bottom-right angles of the rectangle are equal.", caption: none,
)


      
      #solutionbox[
AAS: Two angles and a non-included side in one triangle are congruent to the corresponding angles and side in the second triangle.
]
]


    
    #examplebox("Example 5")[Determining the Congruence Theorem][
What congruence theorem is illustrated in ?


        #figure(
  stdfig([#grid(columns: 2, gutter: 1.5em,
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 191.78pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[Triangle 1])
  line((22.22pt, 22.22pt), (177.78pt, 44.44pt), (88.89pt, 155.56pt), close: true, fill: none, stroke: 0.7pt + black)
})],
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 214.00pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[Triangle 2])
  line((22.22pt, 22.22pt), (177.78pt, 66.67pt), (44.44pt, 177.78pt), close: true, fill: none, stroke: 0.7pt + black)
})]
)]),
  alt: "Two triangles. All three sides of each triangle are of different lengths.", caption: none,
)


      
      #solutionbox[
The SSS theorem.
]
]


    
  
  
    
=== Similarity


    If two triangles have the same angle measurements and are the same shape but differ in size, the two triangles are similar. The lengths of the sides of one triangle will be proportional to the corresponding sides of the second triangle. Note that a single fraction #math.equation(block: false, alt: "the fraction a over b")[$frac(a, b)$] is called a ratio, but two fractions equal to each other is called a proportion, such as #math.equation(block: false, alt: "the fraction a over b equals the fraction c over d .")[$frac(a, b) = frac(c, d) .$]


    This rule of similarity applies to all shapes as well as triangles. Another way to view similarity is by applying a #strong[scaling factor], which is the ratio of corresponding measurements between an object or representation of the object, to an image that produces the second, similar image.


    For example, why are the two images in  are similar? These two images have the same proportions between elements. Therefore, they are similar.


    #figure(figph[Two smiley faces. The first one is bigger and the second one is smaller.], alt: "Two smiley faces. The first one is bigger and the second one is smaller.", caption: [Similarity])


    #examplebox("Example 6")[Determining If Triangles Are Similar][
Are the two triangles shown in  similar?


        #figure(
  stdfig([#grid(columns: 2, gutter: 1.5em,
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 147.33pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle alpha])
  line((22.22pt, 22.22pt), (177.78pt, 22.22pt), (22.22pt, 111.11pt), close: true, fill: none, stroke: 0.7pt + black)
  content((105.93pt, 15.45pt), anchor: "north", text(size: 9pt)[7])
  content((107.81pt, 71.13pt), anchor: "west", text(size: 9pt)[8.06])
  content((13.57pt, 69.14pt), anchor: "east", text(size: 9pt)[4])
  line((22.22pt, 22.22pt), (30.22pt, 22.22pt), (30.22pt, 30.22pt), (22.22pt, 30.22pt), close: true, stroke: 0.6pt + black, fill: none)
  arc((177.78pt, 22.22pt), start: 150.26deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((155.55pt, 28.13pt), text(size: 9pt)[33°])
  arc((22.22pt, 111.11pt), start: 270.00deg, stop: 330.26deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((33.77pt, 91.22pt), text(size: 9pt)[57°])
  content((77.78pt, 55.56pt), anchor: "center", text(size: 9pt)[alpha])
})],
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 147.33pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle beta])
  line((150.00pt, 33.33pt), (150.00pt, 100.00pt), (33.33pt, 100.00pt), close: true, fill: none, stroke: 0.7pt + black)
  content((158.65pt, 64.19pt), anchor: "west", text(size: 9pt)[2])
  content((85.74pt, 106.77pt), anchor: "south", text(size: 9pt)[3.5])
  content((83.85pt, 62.20pt), anchor: "east", text(size: 9pt)[4.03])
  arc((150.00pt, 33.33pt), start: 90.00deg, stop: 150.26deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((138.46pt, 53.23pt), text(size: 9pt)[57°])
  arc((33.33pt, 100.00pt), start: 330.26deg, stop: 0.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((55.56pt, 94.10pt), text(size: 9pt)[33°])
  line((150.00pt, 100.00pt), (142.00pt, 100.00pt), (142.00pt, 92.00pt), (150.00pt, 92.00pt), close: true, stroke: 0.6pt + black, fill: none)
  content((116.67pt, 83.33pt), anchor: "center", text(size: 9pt)[beta])
})]
)]),
  alt: "Two right triangles. In the first triangle, the legs measure 4 and 7. The hypotenuse measures 8.06. The top, bottom-left, and bottom-right angles measure 57 degrees, 90 degrees, and 33 degrees. The triangle is labeled alpha. In the second triangle, the legs measure 3.5 and 2. The hypotenuse measures 4.03. The bottom, top-left, and top-right angles measure 57 degrees, 33 degrees, and 90 degrees. The triangle is labeled beta.", caption: none,
)


      
      #solutionbox[
#strong[Step 1:] We will look at the proportions within each triangle. In triangle #math.equation(block: false, alt: "α")[$α$] (alpha), the side opposite the #math.equation(block: false, alt: "57 degrees")[$57^(∘)$] angle measures 7, and the side opposite the #math.equation(block: false, alt: "33 degrees")[$33^(∘)$] angle measures 4. Then, the measures of the corresponding sides in triangle #math.equation(block: false, alt: "β")[$β$] (beta) measures 3.5 and 2, respectively. We have


        #math.equation(block: true, alt: "the fraction 4 over 7 equals 0.5714 the fraction 2 over 3.5 equals 0.5714.")[$frac(4, 7) = 0.5714 #h(2em) frac(2, 3.5) = 0.5714.$]
        This is the proportion #math.equation(block: false, alt: "the fraction 4 over 7 equals the fraction 2 over 3.5")[$frac(4, 7) = frac(2, 3.5)$]. The scaling factor is 0.5714.


        #strong[Step 2:] Let’s try another correspondence. In triangle #math.equation(block: false, alt: "α")[$α$], the hypotenuse measures 8.06 and the side opposite the #math.equation(block: false, alt: "57 degrees")[$57^(∘)$] angle measures 7. In triangle #math.equation(block: false, alt: "β")[$β$], the hypotenuse measure 4.03 and the side opposite the #math.equation(block: false, alt: "57 degrees")[$57^(∘)$] angle measures 3.5. We have


        #math.equation(block: true, alt: "the fraction 7 over 8.06 equals 0.8685 the fraction 3.5 over 4.03 equals 0.8685.")[$frac(7, 8.06) = 0.8685 #h(2em) frac(3.5, 4.03) = 0.8685.$]
        #strong[Step 3:] Now, let’s look at the proportions between triangle #math.equation(block: false, alt: "α")[$α$] and triangle #math.equation(block: false, alt: "β .")[$β .$] The side measuring 2 in triangle #math.equation(block: false, alt: "β")[$β$] corresponds to the side measuring 4 in triangle #math.equation(block: false, alt: "α")[$α$], the side measuring 3.5 in triangle #math.equation(block: false, alt: "β")[$β$] corresponds to the side measuring 7 in triangle #math.equation(block: false, alt: "α ,")[$α ,$] and the hypotenuse in triangle #math.equation(block: false, alt: "β")[$β$] corresponds to the hypotenuse in triangle #math.equation(block: false, alt: "α .")[$α .$] We have


        #math.equation(block: true, alt: "the fraction 2 over 4 equals 0.5 the fraction 3.5 over 7 equals 0.5 the fraction 4.03 over 8.06 equals 0.5")[$frac(2, 4) = 0.5 #h(2em) frac(3.5, 7) = 0.5 #h(2em) frac(4.03, 8.06) = 0.5$]
        
        Thus, the corresponding angles are equal and the proportions between each pair of corresponding sides equals 0.5. In other words, the scaling factor is 0.5. Therefore, the triangles are similar.
]
]


    
    #examplebox("Example 7")[Proving Similarity][
In , is triangle #math.equation(block: false, alt: "δ")[$δ$] (delta) similar to triangle #math.equation(block: false, alt: "ε")[$ε$] (epsilon)? Find the lengths of sides #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] as part of your answer.


        #figure(
  stdfig([#grid(columns: 2, gutter: 1.5em,
  [#cetz.canvas({
  import cetz.draw: *
  content((71.43pt, 214.00pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle delta])
  line((28.57pt, 28.57pt), (114.29pt, 28.57pt), (28.57pt, 171.43pt), close: true, fill: none, stroke: 0.7pt + black)
  content((74.01pt, 19.95pt), anchor: "north", text(size: 9pt)[2.375])
  content((76.06pt, 107.72pt), anchor: "south", text(size: 9pt)[6])
  content((21.66pt, 105.76pt), anchor: "east", text(size: 9pt)[x])
  line((28.57pt, 28.57pt), (36.57pt, 28.57pt), (36.57pt, 36.57pt), (28.57pt, 36.57pt), close: true, stroke: 0.6pt + black, fill: none)
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  arc((28.57pt, 171.43pt), start: 270.00deg, stop: 300.96deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((28.57pt, 171.43pt), start: 270.00deg, stop: 300.96deg, radius: 17.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((62.86pt, 85.71pt), anchor: "center", text(size: 9pt)[delta])
})],
  [#cetz.canvas({
  import cetz.draw: *
  content((100.00pt, 134.00pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[triangle epsilon])
  line((40.00pt, 40.00pt), (140.00pt, 40.00pt), (40.00pt, 84.00pt), close: true, fill: none, stroke: 0.7pt + black)
  content((96.76pt, 34.05pt), anchor: "west", text(size: 9pt)[2.475])
  content((98.24pt, 65.62pt), anchor: "west", text(size: 9pt)[y])
  content((31.21pt, 63.93pt), anchor: "east", text(size: 9pt)[1.069])
  line((40.00pt, 40.00pt), (48.00pt, 40.00pt), (48.00pt, 48.00pt), (40.00pt, 48.00pt), close: true, stroke: 0.6pt + black, fill: none)
  arc((140.00pt, 40.00pt), start: 156.25deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((40.00pt, 84.00pt), start: 270.00deg, stop: 336.25deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  arc((40.00pt, 84.00pt), start: 270.00deg, stop: 336.25deg, radius: 17.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((80.00pt, 60.00pt), anchor: "center", text(size: 9pt)[epsilon])
})]
)]),
  alt: "Two right triangles are labeled delta and epsilon. In the first triangle, the legs measure x and 2.375. The hypotenuse measures 6. In the second triangle, the legs measure 2.475 and 1.069. The hypotenuse measures y. The angles in both the triangles are congruent.", caption: none,
)


      
      #solutionbox[
We can see that all three angles in triangle #math.equation(block: false, alt: "δ")[$δ$] are equal to the corresponding angles in triangle #math.equation(block: false, alt: "ε")[$ε$]. That is enough to determine similarity. However, we want to find the values of #math.equation(block: false, alt: "x")[$x$] and #math.equation(block: false, alt: "y")[$y$] to prove similarity.


        #strong[Step 1:] We have to do is set up the proportions between the corresponding sides. We have the side that measures 2.375 in triangle #math.equation(block: false, alt: "δ")[$δ$] corresponding to the side measuring 1.069 in triangle #math.equation(block: false, alt: "ε")[$ε$]. We have the hypotenuse/side in triangle #math.equation(block: false, alt: "δ")[$δ$] measuring 6 corresponding to the hypotenuse/side labeled #math.equation(block: false, alt: "y")[$y$] in triangle #math.equation(block: false, alt: "ε")[$ε$]. And, finally, the side labeled #math.equation(block: false, alt: "x")[$x$] in triangle #math.equation(block: false, alt: "δ")[$δ$] corresponds to the side measuring 2.475 in triangle #math.equation(block: false, alt: "ε .")[$ε .$]


        Each proportion should be equal. We start with the proportion of the shorter sides. Thus


        #math.equation(block: true, alt: "the fraction 1.069 over 2.375 equals 0.45")[$frac(1.069, 2.375) = 0.45$]
        #strong[Step 2:] We solve for #math.equation(block: false, alt: "y")[$y$] using the first proportion. Set the two ratios equal to each other, cross-multiply, and solve for #math.equation(block: false, alt: "y")[$y$]. We have:


        #math.equation(block: true, alt: "the fraction 1.069 over 2.375, equals, the fraction y over 6; open parenthesis 6 close parenthesis open parenthesis 1.069 close parenthesis, equals, open parenthesis 2.375 close parenthesis open parenthesis y close parenthesis; 6.414, equals, 2.375 y; the fraction 6.414 over 2.375, equals, 2.7 equals y")[$frac(1.069, 2.375) & = & frac(y, 6) \ ( 6 ) ( 1.069 ) & = & ( 2.375 ) ( y ) \ 6.414 & = & 2.375 y \ frac(6.414, 2.375) & = & 2.7 = y$]
        So, #math.equation(block: false, alt: "y equals 2.7")[$y = 2.7$].


        #strong[Step 3:] Checking that length in the proportion factor of 0.45, we have:


        #math.equation(block: true, alt: "the fraction y over 6 equals the fraction 2.7 over 6 equals 0.45")[$frac(y, 6) = frac(2.7, 6) = 0.45$]
        #strong[Step 4:] Solving for #math.equation(block: false, alt: "x")[$x$], we will use the same proportion we used to solve for #math.equation(block: false, alt: "y")[$y$]. We have:


        #math.equation(block: true, alt: "the fraction 1.069 over 2.375, equals, the fraction 2.475 over x; 1.069 open parenthesis x close parenthesis, equals, 2.475 open parenthesis 2.375 close parenthesis; 1.069 open parenthesis x close parenthesis, equals, 5.878; x, equals, the fraction 5.878 over 1.069 equals 5.5; the fraction 2.475 over x, equals, the fraction 2.475 over 5.5 equals 0.45")[$frac(1.069, 2.375) & = & frac(2.475, x) \ 1.069 ( x ) & = & 2.475 ( 2.375 ) \ 1.069 ( x ) & = & 5.878 \ x & = & frac(5.878, 1.069) = 5.5 \ frac(2.475, x) & = & frac(2.475, 5.5) = 0.45$]
        #strong[Step 5:] We test the proportions. We have the following:


        #math.equation(block: true, alt: "the fraction 2.7 over 6 equals the fraction 2.475 over 5.5 equals the fraction 1.069 over 2.375 equals 0.45")[$frac(2.7, 6) = frac(2.475, 5.5) = frac(1.069, 2.375) = 0.45$]
        
        The proportions are all equal. Therefore, we have proven the property of similarity between triangle #math.equation(block: false, alt: "δ")[$δ$] and triangle #math.equation(block: false, alt: "ε .")[$ε .$]
]
]


    
    #examplebox("Example 8")[Applying Similar Triangles][
A person who is 5 feet tall is standing 50 feet away from the base of a tree . The tree casts a 57-foot shadow. The person casts a 7-foot shadow. What is the height of the tree?


        #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  line((3.39pt, 3.39pt), (3.39pt, 30.51pt), stroke: 0.7pt + black)
  content((-5.61pt, 16.95pt), anchor: "south", text(size: 9pt)[x])
  line((3.39pt, 3.39pt), (172.88pt, 3.39pt), stroke: 0.7pt + black)
  content((88.14pt, 12.39pt), anchor: "south", text(size: 9pt)[50 ft])
  line((172.88pt, 3.39pt), (196.61pt, 3.39pt), stroke: 0.7pt + black)
  content((184.75pt, 12.39pt), anchor: "south", text(size: 9pt)[7 ft])
  line((172.88pt, 3.39pt), (172.88pt, 20.34pt), stroke: 0.7pt + black)
  content((163.88pt, 11.86pt), anchor: "south", text(size: 9pt)[5 ft])
  line((3.39pt, 30.51pt), (196.61pt, 3.39pt), stroke: 0.7pt + black)
  content((101.25pt, 25.86pt), anchor: "south", text(size: 9pt)[unknown])
  line((3.39pt, 3.39pt), (11.39pt, 3.39pt), (11.39pt, 11.39pt), (3.39pt, 11.39pt), close: true, stroke: 0.6pt + black, fill: none)
})]),
  alt: "An illustration of a palm tree and stick figure person shows a right triangle. The vertical leg shows a tree and it measures x. The hypotenuse is unknown. The horizontal leg measures 57 feet. A boy who is 5 feet tall is standing 50 feet away from the base of the tree. The boy casts a shadow of 7 feet.", caption: none,
)


      
      #solutionbox[
The bigger triangle includes a tree at side #math.equation(block: false, alt: "x")[$x$] and the smaller triangle includes the person at the side labeled 5 ft. These two triangles are similar because the smaller triangle fits inside the larger triangle at the smallest angle. It would fit inside the larger triangle at either of the other two angles as well. That all angles are equal is one of the criteria for similar triangles, so we can solve using proportions:


        #math.equation(block: true, alt: "the fraction 5 over 7, equals, the fraction x over 57; 5 open parenthesis 57 close parenthesis, equals, 7 x; the fraction 285 over 7, equals, x equals 40.7")[$frac(5, 7) & = & frac(x, 57) \ 5 ( 57 ) & = & 7 x \ frac(285, 7) & = & x = 40.7$]
        The tree is 40.7 feet tall.
]
]


    
    #examplebox("Example 9")[Finding Missing Lengths][
At a certain time of day, a radio tower casts a shadow 180 feet long . At the same time, a 9-foot truck casts a shadow 15 feet long. What is the height of the tower?


        #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  line((1.10pt, 1.10pt), (198.90pt, 1.10pt), (1.10pt, 110.99pt), close: true, fill: none, stroke: 0.7pt + black)
  content((106.02pt, -5.59pt), anchor: "north", text(size: 9pt)[180])
  content((-7.57pt, 58.45pt), anchor: "east", text(size: 9pt)[x])
  line((165.93pt, 1.10pt), (182.42pt, 1.10pt), (182.42pt, 10.99pt), close: true, fill: none, stroke: 0.7pt + black)
  content((168.41pt, -5.82pt), anchor: "north", text(size: 9pt)[15])
  content((191.04pt, 8.63pt), anchor: "west", text(size: 9pt)[9])
  line((1.10pt, 1.10pt), (9.10pt, 1.10pt), (9.10pt, 9.10pt), (1.10pt, 9.10pt), close: true, stroke: 0.6pt + black, fill: none)
  line((182.42pt, 1.10pt), (174.42pt, 1.10pt), (174.42pt, 9.10pt), (182.42pt, 9.10pt), close: true, stroke: 0.6pt + black, fill: none)
})]),
  alt: "An illustration shows a right triangle. The vertical leg resembles a tower of x feet high. The horizontal leg measures 180 feet. A truck of 9 feet casts a shadow of 15 feet. The truck lies 15 feet to the left from the bottom-right of the triangle.", caption: none,
)


      
      #solutionbox[
These are similar triangles and the problem can be solved by using proportions:


        #math.equation(block: true, alt: "the fraction x over 180, equals, the fraction 9 over 15; 9 open parenthesis 180 close parenthesis, equals, 15 x; 1620, equals, 15 x; 108, equals, x")[$frac(x, 180) & = & frac(9, 15) \ 9 ( 180 ) & = & 15 x \ 1620 & = & 15 x \ 108 & = & x$]
        The height of the tower is 108 ft.
]
]


    
    #notebox("People in Mathematics", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[
#emph[Thales of Miletus]


      Thales of Miletus, sixth century BC, is considered one of the greatest mathematicians and philosophers of all time. Thales is credited with being the first to discover that the two angles at the base of an isosceles triangle are equal, and that the two angles formed by intersecting lines are equal—that is, vertical or opposite angles, are equal. Thales is also known for devising a method for measuring the height of the pyramids by similar right triangles.  shows his method. He measured the length of the shadow cast by the pyramid at the precise time when his own shadow ended at the same place.


      #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  line((86.35pt, 96.00pt), (24.00pt, 60.00pt), stroke: (paint: black, thickness: 0.6pt, dash: "dashed"))
  line((24.00pt, 60.00pt), (86.35pt, 156.00pt), stroke: (paint: black, thickness: 0.6pt, dash: "dashed"))
  line((24.00pt, 60.00pt), (86.35pt, 24.00pt), stroke: 0.7pt + black)
  line((86.35pt, 24.00pt), (148.71pt, 60.00pt), stroke: 0.7pt + black)
  line((148.71pt, 60.00pt), (86.35pt, 96.00pt), stroke: 0.7pt + black)
  line((86.35pt, 24.00pt), (86.35pt, 156.00pt), stroke: 0.7pt + black)
  line((148.71pt, 60.00pt), (86.35pt, 156.00pt), stroke: 0.7pt + black)
  line((86.35pt, 96.00pt), (86.35pt, 156.00pt), stroke: 0.7pt + black)
  line((86.35pt, 60.00pt), (86.35pt, 156.00pt), stroke: (paint: black, thickness: 0.6pt, dash: "dashed"))
  content((78.35pt, 108.00pt), anchor: "east", text(size: 9pt)[B′])
  content((117.53pt, 34.00pt), anchor: "north", text(size: 9pt)[A′])
  content((125.53pt, 108.00pt), anchor: "west", text(size: 9pt)[C′])
  content((117.53pt, 34.00pt), anchor: "north", text(size: 9pt)[A])
  content((125.53pt, 108.00pt), anchor: "west", text(size: 9pt)[C])
})]),
  alt: "An illustration shows a right triangle. The vertical leg represents the height of the pyramid and it measures B prime. Sun is on the right. A vertical line, B is to the left of the pyramid. The horizontal leg measures A prime and the hypotenuse measures C prime. The distance from B to the bottom-left vertex along the horizontal leg is marked A. The distance from B to the bottom-left vertex along the hypotenuse is marked C.", caption: [Thales and Similarity],
)


      He equated the vertical height of the pyramid with his own height; the horizontal distance from the pyramid to the tip of its shadow with the distance from himself and the tip of his own shadow; and finally, the length of the shadow cast off the top of the pyramid with length of his own shadow cast off the top of his head. Using proportions, as shown in , he essentially discovered the properties of similarity for right triangles. That is, #math.equation(block: false, alt: "△ A B C")[$△ A B C$] is similar to #math.equation(block: false, alt: "△ A prime B prime C prime .")[$△ A^(′) B^(′) C^(′) .$] Note that to be similar, all corresponding angles between the two triangles must be equal, and the proportions from one side to another side within each triangle, as well as the proportions of the corresponding sides between the two triangles must be equal.


      Thales is also credited with discovering a method of determining the distance of a ship from the shoreline. Here is how he did it, as illustrated in .


      #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
  content((80.00pt, 214.00pt), text(font: sys.inputs.at("title-font", default: "STIX Two Text"), size: 11pt)[Triangles ABC and CDE])
  line((140.00pt, 180.00pt), (20.00pt, 180.00pt), (80.00pt, 100.00pt), close: true, fill: none, stroke: 0.7pt + black)
  content((80.00pt, 189.00pt), anchor: "south", text(size: 9pt)[x])
  content((41.78pt, 136.34pt), anchor: "east", text(size: 9pt)[60])
  line((80.00pt, 100.00pt), (20.00pt, 20.00pt), (140.00pt, 20.00pt), close: true, fill: none, stroke: 0.7pt + black)
  content((80.00pt, 11.00pt), anchor: "north", text(size: 9pt)[330])
  content((118.22pt, 63.66pt), anchor: "west", text(size: 9pt)[45])
  content((136.00pt, 176.00pt), anchor: "north-east", text(size: 9pt)[A])
  content((24.00pt, 176.00pt), anchor: "north-west", text(size: 9pt)[B])
  content((76.00pt, 100.00pt), anchor: "east", text(size: 9pt)[C])
  content((136.00pt, 24.00pt), anchor: "south-east", text(size: 9pt)[D])
  content((24.00pt, 24.00pt), anchor: "south-west", text(size: 9pt)[E])
  content((140.00pt, 194.00pt), anchor: "south", text(size: 9pt)[ship])
  content((20.00pt, 194.00pt), anchor: "south", text(size: 9pt)[shoreline])
})]),
  alt: "Two triangles, A B C and C D E. The two triangles share the same point C. The second triangle is inverted. Point A is labeled ship. Point B is labeled shoreline. A B measures x. B C measures 60 feet. C D measures 45 feet. D E measures 330 feet.", caption: [Thales and Similar Triangles],
)


      Thales walked along the shoreline pointing a stick at the ship until it formed a #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] angle to the shore. Then he walked along the shot and placed the stick in the ground at point #math.equation(block: false, alt: "C")[$C$]. He continued walking until he reached point #math.equation(block: false, alt: "D")[$D$]. Then, he turned and walked away from the shore at a #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] angle until the stick he placed in the ground at point #math.equation(block: false, alt: "C")[$C$] lined up with the ship, point #math.equation(block: false, alt: "E")[$E$]. This is how he created similar triangles and estimated the distance of the ship to the shore by using proportions.
]


  
  
  
  
    
=== Key Terms


    - acute
- obtuse
- isosceles
- equilateral
- hypotenuse
- congruence
- similarity
- scaling factor


  
  
    
=== Key Concepts


    - The sum of the interior angles of a triangle equals #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$]
- Two triangles are congruent when the corresponding angles have the same measure and the corresponding side lengths are equal.
- The congruence theorems include the following: SAS, two sides and the included angle of one triangle are congruent to the same in a second triangle; ASA, two angles and the included side of one triangle are congruent to the same in a second triangle; SSS, all three side lengths of one triangle are congruent to the same in a second triangle; AAS, two angles and the non-included side of one triangle are congruent to the same in a second triangle.
- Two shapes are similar when the proportions between corresponding angles, sides or features of two shapes are equal, regardless of size.