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== 10.2#h(0.6em)Angles


  #figure(figph[The exterior view of an architectural building.], alt: "The exterior view of an architectural building.", caption: [This modern architectural design emphasizes sharp reflective angles as part of the aesthetic through the use of glass walls.])


  
    
=== Learning Objectives


    After completing this section, you should be able to:


    + Identify and express angles using proper notation.
+ Classify angles by their measurement.
+ Solve application problems involving angles.
+ Compute angles formed by transversals to parallel lines.
+ Solve application problems involving angles formed by parallel lines.


  
  Unusual perspectives on architecture can reveal some extremely creative images. For example, aerial views of cities reveal some exciting and unexpected angles. Add reflections on glass or steel, lighting, and impressive textures, and the structure is a work of art. Understanding angles is critical to many fields, including engineering, architecture, landscaping, space planning, and so on. This is the topic of this section.


  We begin our study of angles with a description of how angles are formed and how they are classified. An angle is the joining of two rays, which sweep out as the sides of the angle, with a common endpoint. The common endpoint is called the #strong[vertex]. We will often need to refer to more than one vertex, so you will want to know the plural of vertex, which is vertices.


  In , let the ray #math.equation(block: false, alt: "A B →")[$arrow(A B)$] stay put. Rotate the second ray #math.equation(block: false, alt: "A C →")[$arrow(A C)$] in a counterclockwise direction to the size of the angle you want. The angle is formed by the amount of rotation of the second ray. When the ray #math.equation(block: false, alt: "A C →")[$arrow(A C)$] continues to rotate in a counterclockwise direction back to its original position coinciding with ray #math.equation(block: false, alt: "A B → ,")[$arrow(A B) ,$] the ray will have swept out #math.equation(block: false, alt: "360 degrees .")[$360^(∘) .$] We call the rays the “sides” of the angle.


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=== Classifying Angles


    Angles are measured in #strong[radians] or #strong[degrees]. For example, an angle that measures #math.equation(block: false, alt: "π")[$π$] radians, or 3.14159 radians, is equal to the angle measuring #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$] An angle measuring #math.equation(block: false, alt: "the fraction π over 2")[$frac(π, 2)$] radians, or 1.570796 radians, measures #math.equation(block: false, alt: "90 degrees .")[$90^(∘) .$] To translate degrees to radians, we multiply the angle measure in degrees by #math.equation(block: false, alt: "the fraction π over 180 .")[$frac(π, 180) .$] For example, to write #math.equation(block: false, alt: "45 degrees")[$45^(∘)$] in radians, we have


    #math.equation(block: true, alt: "45 degrees open parenthesis the fraction π over 180 close parenthesis equals the fraction π over 4 equals 0.785398 radians.")[$45^(∘) ( frac(π, 180) ) = frac(π, 4) = 0.785398 #h(0.28em) "radians."$]
    To translate radians to degrees, we multiply by #math.equation(block: false, alt: "the fraction 180 over π .")[$frac(180, π) .$] For example, to write #math.equation(block: false, alt: "2 π")[$2 π$] radians in degrees, we have


    #math.equation(block: true, alt: "2 π open parenthesis the fraction 180 over π close parenthesis equals 360 degrees .")[$2 π ( frac(180, π) ) = 360^(∘) .$]
    Another example of translating radians to degrees and degrees to radians is #math.equation(block: false, alt: "the fraction 2 π over 3 .")[$frac(2 π, 3) .$] To write in degrees, we have #math.equation(block: false, alt: "the fraction 2 π over 3 open parenthesis the fraction 180 over π close parenthesis equals 120 degrees .")[$frac(2 π, 3) ( frac(180, π) ) = 120^(∘) .$] To write #math.equation(block: false, alt: "30 degrees")[$30^(∘)$] in radians, we have #math.equation(block: false, alt: "30 degrees open parenthesis the fraction π over 180 close parenthesis equals the fraction π over 6")[$30^(∘) ( frac(π, 180) ) = frac(π, 6)$]. However, we will use degrees throughout this chapter.


    #notebox("Formula", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[
To translate an angle measured in degrees to radians, multiply by #math.equation(block: false, alt: "the fraction π over 180 .")[$frac(π, 180) "."$]


      To translate an angle measured in radians to degrees, multiply by #math.equation(block: false, alt: "the fraction 180 over π .")[$frac(180, π) .$]
]


    Several angles are referred to so often that they have been given special names. A #strong[straight angle] measures #math.equation(block: false, alt: "180 degrees")[$180^(∘)$]; a #strong[right angle] measures #math.equation(block: false, alt: "90 degrees ;")[$90^(∘) ;$] an #strong[acute angle] is any angle whose measure is less than #math.equation(block: false, alt: "90 degrees ;")[$90^(∘) ;$] and an #strong[obtuse angle] is any angle whose measure is between #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] and #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$]


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    An easy way to measure angles is with a protractor . A protractor is a very handy little tool, usually made of transparent plastic, like the one shown here.


    #figure(figph[A protractor with its center labeled and an inch ruler is across the bottom.], alt: "A protractor with its center labeled and an inch ruler is across the bottom.", caption: [Protractor])


    With a protractor, you line up the straight bottom with the horizontal straight line of the angle. Be sure to have the center hole lined up with the vertex of the angle. Then, look for the mark on the protractor where the second ray lines up. As you can see from the image, the degrees are marked off. Where the second ray lines up is the measurement of the angle.


    #notebox("Checkpoint", rgb("#059669"), rgb("#007942"), rgb("#EAF3EC"))[
#emph[Make sure you correctly match the center mark of the protractor with the vertex of the angle to be measured. Otherwise, you will not get the correct measurement. Also, keep the protractor in a vertical position.]
]


  
  
    
=== Notation


    Naming angles can be done in couple of ways. We can name the angle by three points, one point on each of the sides and the vertex point in the middle, or we can name it by the vertex point alone. Also, we can use the symbols #math.equation(block: false, alt: "∠")[$∠$] or #math.equation(block: false, alt: "∡")[$∡$] before the points. When we are referring to the measure of the angle, we use the symbol #math.equation(block: false, alt: "m ∡")[$m ∡$].


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    We can name this angle #math.equation(block: false, alt: "∡ B A C")[$∡ B A C$], or #math.equation(block: false, alt: "∡ C A B")[$∡ C A B$], or #math.equation(block: false, alt: "∡ A .")[$∡ A .$]


    #examplebox("Example 1")[Classifying Angles][
Determine which angles are acute, right, obtuse, or straight on the graph . You may want to use a protractor for this one.


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#figure(table(
  columns: 4, align: left, inset: 6pt,
  table.header([Acute angles measure less than #math.equation(block: false, alt: "90 degrees .")[$90^(∘) .$]], [Obtuse angles measure between #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] and #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$]], [Right angles measure #math.equation(block: false, alt: "90 degrees .")[$90^(∘) .$]], [Straight angles measure #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$]]),
  [#math.equation(block: false, alt: "∠ E O F; ∠ E O G; ∠ F O G; ∠ G O H; ∠ F O H; ∠ H O J; ∠ H O K; ∠ J O K; ∠ K O L; ∠ J O L")[$∠ E O F \ ∠ E O G \ ∠ F O G \ ∠ G O H \ ∠ F O H \ ∠ H O J \ ∠ H O K \ ∠ J O K \ ∠ K O L \ ∠ J O L$]], [#math.equation(block: false, alt: "∠ E O J; ∠ E O K; ∠ F O J; ∠ F O K; ∠ F O L; ∠ G O K; ∠ G O L")[$∠ E O J \ ∠ E O K \ ∠ F O J \ ∠ F O K \ ∠ F O L \ ∠ G O K \ ∠ G O L$]], [#math.equation(block: false, alt: "∠ E O H; ∠ H O L; ∠ G O J")[$∠ E O H \ ∠ H O L \ ∠ G O J$]], [#math.equation(block: false, alt: "∠ E O L")[$∠ E O L$]],
))


        Most angles can be classified visually or by description. However, if you are unsure, use a protractor.
]
]


    
  
  
    
=== Adjacent Angles


    Two angles with the same starting point or vertex and one common side are called adjacent angles. In , angle #math.equation(block: false, alt: "∠ D B C")[$∠ D B C$] is adjacent to #math.equation(block: false, alt: "∠ C B A")[$∠ C B A$]. Notice that the way we designate an angle is with a point on each of its two sides and the vertex in the middle.


    #figure(
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  import cetz.draw: *
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  content((162.67pt, 37.33pt), anchor: "south-east", text(size: 9pt)[D])
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  alt: "Three rays, B A, B C, and B D originate from the same point, B. The rays, B A, and B C make an acute angle. The rays, B C, and B D make an acute angle. The angle, A B D is acute.", caption: [Adjacent Angles],
)


  
  
    
=== Supplementary Angles


    Two angles are #strong[supplementary] if the sum of their measures equals #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$] In , we are given that #math.equation(block: false, alt: "m ∡ F B E equals 35 degrees ,")[$m ∡ F B E = 35^(∘) ,$] so what is #math.equation(block: false, alt: "m ∡ A B E ?")[$m ∡ A B E ?$] These are supplementary angles. Therefore, because #math.equation(block: false, alt: "m ∡ A B F equals 180 degrees")[$m ∡ A B F = 180^(∘)$], and as #math.equation(block: false, alt: "180 degrees minus 35 degrees equals 145 degrees ,")[$180^(∘) − 35^(∘) = 145^(∘) ,$] we have #math.equation(block: false, alt: "m ∡ A B E equals 145 degrees .")[$m ∡ A B E = 145^(∘) .$]


    #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
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  circle((60.00pt, 80.00pt), radius: 1.8pt, fill: black, stroke: none)
  content((64.00pt, 76.00pt), anchor: "north-west", text(size: 9pt)[E])
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  content((156.00pt, 56.00pt), anchor: "north-east", text(size: 9pt)[C])
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  alt: "Five rays originate from the same point, B. The rays, B F, and B A are horizontal. The ray, B D is vertical. The ray, B E lies between B F and B D and it makes an acute angle with each ray. The ray, B C lies between B D and B A, and it makes an acute angle with each ray.", caption: [Supplementary Angles],
)


    #examplebox("Example 2")[Solving for Angle Measurements and Supplementary Angles][
Solve for the angle measurements in .


        #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
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  arc((100.00pt, 20.00pt), start: 33.69deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((93.33pt, 42.01pt), text(size: 9pt)[32x - 7])
})]),
  alt: "A horizontal line with a ray originating from its center. The line makes an acute angle, 5 x plus 2 with the ray, and an obtuse angle, 32 x minus 7 with the ray.", caption: none,
)


      
      #solutionbox[
#strong[Step 1:] These are supplementary angles. We can see this because the two angles are part of a horizontal line, and a horizontal line represents #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$] Therefore, the sum of the two angles equals #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$]


        #strong[Step 2:]


        #math.equation(block: true, alt: "open parenthesis 32 x minus 7 close parenthesis plus open parenthesis 5 x plus 2 close parenthesis, equals, 180; 37 x minus 5, equals, 180; 37 x, equals, 185; x, equals, 5")[$( 32 x − 7 ) + ( 5 x + 2 ) & = & 180 \ 37 x − 5 & = & 180 \ 37 x & = & 185 \ x & = & 5$]
        #strong[Step 3:] Find the measure of each angle:


        #math.equation(block: true, alt: "32 x minus 7, equals, 32 open parenthesis 5 close parenthesis minus 7; equals, 153 degrees; 5 x plus 2, equals, 5 open parenthesis 5 close parenthesis plus 2; equals, 27 degrees")[$32 x − 7 & = & 32 ( 5 ) − 7 \  & = & 153^(∘) \ 5 x + 2 & = & 5 ( 5 ) + 2 \  & = & 27^(∘)$]
        #strong[Step 4:] We check: #math.equation(block: false, alt: "153 degrees plus 27 degrees equals 180 degrees .")[$153^(∘) + 27^(∘) = 180^(∘) .$]
]
]


    
  
  
    
=== Complementary Angles


    Two angles are #strong[complementary] if the sum of their measures equals #math.equation(block: false, alt: "90 degrees .")[$90^(∘) .$] In , we have #math.equation(block: false, alt: "m ∡ A B C equals 30 degrees ,")[$m ∡ A B C = 30^(∘) ,$] and #math.equation(block: false, alt: "m ∡ A B D equals 90 degrees .")[$m ∡ A B D = 90^(∘) .$] What is the #math.equation(block: false, alt: "m ∡ C B D ?")[$m ∡ C B D ?$] These are complementary angles. Therefore, because #math.equation(block: false, alt: "90 degrees minus 30 degrees equals 60 degrees ,")[$90^(∘) − 30^(∘) = 60^(∘) ,$] the #math.equation(block: false, alt: "∡ C B D equals 60 degrees .")[$∡ C B D = 60^(°) .$]


    #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
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  content((84.00pt, 176.00pt), anchor: "north-west", text(size: 9pt)[D])
})]),
  alt: "Two lines, A B and B D intersect each other forming a right angle. A ray, B C makes an acute angle, 6 x minus 5 with the line, B A. Another ray originating from B makes an acute angle, 9 x with the line, B D. This ray and B C make an acute angle of 20 degrees.", caption: [Complementary Angles],
)


    #examplebox("Example 3")[Solving for Angle Measurements and Complementary Angles][
Solve for the angle measurements in .


        #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
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})]),
  alt: "Two lines intersect each other forming a right angle. A ray makes an acute angle, 7 x minus 5 with the horizontal line. Another ray originating from the intersection point of the lines makes an acute angle, 9 x minus 5 with the vertical line. An acute angle of 4 x is formed by these two rays.", caption: none,
)


      
      #solutionbox[
We have that


        #math.equation(block: true, alt: "open parenthesis 9 x minus 5 close parenthesis plus 4 x plus open parenthesis 7 x minus 5 close parenthesis, equals, 90; 20 x, equals, 100; x, equals, 5")[$( 9 x − 5 ) + 4 x + ( 7 x − 5 ) & = & 90 \ 20 x & = & 100 \ x & = & 5$]
        Then, #math.equation(block: false, alt: "m ∡ open parenthesis 9 x minus 5 close parenthesis equals 40 degrees")[$m ∡ ( 9 x − 5 ) = 40^(∘)$], #math.equation(block: false, alt: "m ∡ open parenthesis 4 x close parenthesis equals 20 degrees")[$m ∡ ( 4 x ) = 20^(∘)$], and #math.equation(block: false, alt: "m ∡ open parenthesis 7 x minus 5 close parenthesis equals 30 degrees .")[$m ∡ ( 7 x − 5 ) = 30^(∘) .$]
]
]


    
  
  
    
=== Vertical Angles


    When two lines intersect, the opposite angles are called vertical angles, and vertical angles have equal measure. For example,  shows two straight lines intersecting each other. One set of opposite angles shows angle markers; those angles have the same measure. The other two opposite angles have the same measure as well.


    #figure(figph[Two lines intersect each other. One set of opposite angles is shaded.], alt: "Two lines intersect each other. One set of opposite angles is shaded.", caption: [Vertical Angles])


    #examplebox("Example 4")[Calculating Vertical Angles][
In , one angle measures #math.equation(block: false, alt: "40 degrees .")[$40^(∘) .$] Find the measures of the remaining angles.


        #figure(
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  content((107.27pt, 44.85pt), text(size: 9pt)[3])
})]),
  alt: "Two lines intersect each other. One set of opposite angles is labeled 1 and 3. The other set of opposite angles is labeled 2 and 40 degrees.", caption: none,
)


      
      #solutionbox[
The 40-degree angle and #math.equation(block: false, alt: "∠ 2")[$∠ 2$] are vertical angles. Therefore, #math.equation(block: false, alt: "m ∡ 2 equals 40 degrees .")[$m ∡ 2 = 40^(∘) .$]


        Notice that #math.equation(block: false, alt: "∡ 2")[$∡ 2$] and #math.equation(block: false, alt: "∡ 1")[$∡ 1$] are supplementary angles, meaning that the sum of #math.equation(block: false, alt: "m ∡ 2")[$m ∡ 2$] and #math.equation(block: false, alt: "m ∡ 1")[$m ∡ 1$] equals #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$] Therefore, #math.equation(block: false, alt: "m ∡ 1 equals 180 degrees minus 40 degrees equals 140 degrees")[$m ∡ 1 = 180^(∘) − 40^(∘) = 140^(∘)$].


        Since #math.equation(block: false, alt: "∡ 1")[$∡ 1$] and #math.equation(block: false, alt: "∡ 3")[$∡ 3$] are vertical angles, then #math.equation(block: false, alt: "m ∡ 3")[$m ∡ 3$] equals #math.equation(block: false, alt: "140 degrees .")[$140^(∘) .$]
]
]


    
  
  
    
=== Transversals


    When two parallel lines are crossed by a straight line or #strong[transversal], eight angles are formed, including alternate interior angles, alternate exterior angles, corresponding angles, vertical angles, and supplementary angles. Angles 1, 2, 7, and 8 are called exterior angles, and angles 3, 4, 5, and 6 are called interior angles.


    #figure(
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  content((186.67pt, 116.67pt), anchor: "west", text(size: 9pt)[l₁])
  content((186.67pt, 50.00pt), anchor: "west", text(size: 9pt)[l₂])
})]),
  alt: "Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles.", caption: [Transversal],
)


  
  
    
=== Alternate Interior Angles


    Alternate interior angles are the interior angles on opposite sides of the transversal. These two angles have the same measure. For example, #math.equation(block: false, alt: "∡ 3")[$∡ 3$] and #math.equation(block: false, alt: "∡ 6")[$∡ 6$] are alternate interior angles and have equal measure; #math.equation(block: false, alt: "∡ 4")[$∡ 4$] and #math.equation(block: false, alt: "∡ 5")[$∡ 5$] are alternate interior angles and have equal measure as well.


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  arc((120.00pt, 40.00pt), start: 135.00deg, stop: 180.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
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  arc((120.00pt, 40.00pt), start: 0.00deg, stop: 135.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
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  content((129.95pt, 64.02pt), text(size: 9pt)[6])
  arc((120.00pt, 40.00pt), start: 180.00deg, stop: 315.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((111.20pt, 18.75pt), text(size: 9pt)[7])
  arc((120.00pt, 40.00pt), start: 315.00deg, stop: 0.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((141.25pt, 31.20pt), text(size: 9pt)[8])
})]),
  alt: "Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The alternate interior angles, 3 and 6 are highlighted.", caption: [Alternate Interior Angles],
)


  
  
    
=== Alternate Exterior Angles


    Alternate exterior angles are exterior angles on opposite sides of the transversal and have the same measure. For example, in , #math.equation(block: false, alt: "∡ 2")[$∡ 2$] and #math.equation(block: false, alt: "∡ 7")[$∡ 7$] are alternate exterior angles and have equal measures; #math.equation(block: false, alt: "∡ 1")[$∡ 1$] and #math.equation(block: false, alt: "∡ 8")[$∡ 8$] are alternate exterior angles and have equal measures as well.


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  arc((140.00pt, 40.00pt), start: 323.13deg, stop: 0.00deg, radius: 14.00pt, anchor: "origin", stroke: 0.6pt + black)
  content((161.82pt, 32.73pt), text(size: 9pt)[8])
})]),
  alt: "Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The alternate exterior angles, 2 and 7 are highlighted.", caption: [Alternate Exterior Angles],
)


  
  
    
=== Corresponding Angles


    Corresponding angles refer to one exterior angle and one interior angle on the same side as the transversal, which have equal measures. In , #math.equation(block: false, alt: "∡ 1")[$∡ 1$] and #math.equation(block: false, alt: "∡ 5")[$∡ 5$] are corresponding angles and have equal measures; #math.equation(block: false, alt: "∡ 3")[$∡ 3$] and #math.equation(block: false, alt: "∡ 7")[$∡ 7$] are corresponding angles and have equal measures; #math.equation(block: false, alt: "∡ 2")[$∡ 2$] and #math.equation(block: false, alt: "∡ 6")[$∡ 6$] are corresponding angles and have equal measures; #math.equation(block: false, alt: "∡ 4")[$∡ 4$] and #math.equation(block: false, alt: "∡ 8")[$∡ 8$] are corresponding angles and have equal measures as well.


    #figure(
  stdfig([#cetz.canvas({
  import cetz.draw: *
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  content((131.20pt, 18.75pt), text(size: 9pt)[8])
})]),
  alt: "Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.", caption: [Corresponding Angles],
)


    #examplebox("Example 5")[Evaluating Space][
You live on the corner of First Avenue and Linton Street. You want to plant a garden in the far corner of your property  and fence off the area. However, the corner of your property does not form the traditional right angle. You learned from the city that the streets cross at an angle equal to #math.equation(block: false, alt: "150 degrees .")[$150^(∘) .$] What is the measure of the angle that will border your garden?


        #figure(figph[Two streets, First Avenue and Linton Street intersect each other. One set of opposite angles is unknown and 150 degrees. The other set of opposite angles shows the garden on the left and blank on the right.], alt: "Two streets, First Avenue and Linton Street intersect each other. One set of opposite angles is unknown and 150 degrees. The other set of opposite angles shows the garden on the left and blank on the right.", caption: none)


      
      #solutionbox[
As the angle between Linton Street and First Avenue is #math.equation(block: false, alt: "150 degrees ,")[$150^(∘) ,$] the supplementary angle is #math.equation(block: false, alt: "30 degrees .")[$30^(∘) .$] Therefore, the garden will form a #math.equation(block: false, alt: "30 degrees")[$30^(∘)$] angle at the corner of your property.
]
]


    
    #examplebox("Example 6")[Determining Angles Formed by a Transversal][
In  given that angle 3 measures #math.equation(block: false, alt: "40 degrees ,")[$40^(∘) ,$] find the measures of the remaining angles and give a reason for your solution.


        #figure(
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  import cetz.draw: *
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  content((190.00pt, 46.00pt), anchor: "center", text(size: 9pt)[l₂])
})]),
  alt: "Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 2, 40 degrees, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 2, 7, and 8 are exterior angles. 40 degrees, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.", caption: none,
)


      
      #solutionbox[
#math.equation(block: false, alt: "m ∡ 2 equals m ∡ 3 equals 40 degrees")[$m ∡ 2 = m ∡ 3 = 40^(∘)$] by vertical angles.


        #math.equation(block: false, alt: "∡ 3 equals m ∡ 7")[$∡ 3 = m ∡ 7$] by corresponding angles.


        #math.equation(block: false, alt: "m ∡ 7 equals m ∡ 6 equals 40 degrees")[$m ∡ 7 = m ∡ 6 = 40^(∘)$] by vertical angles.


        #math.equation(block: false, alt: "m ∡ 1 equals 180 minus 40 equals 140 degrees")[$m ∡ 1 = 180 − 40 = 140^(∘)$] by supplementary angles.


        #math.equation(block: false, alt: "m ∡ 4 equals m ∡ 1 equals 140 degrees")[$m ∡ 4 = m ∡ 1 = 140^(∘)$] by vertical angles.


        #math.equation(block: false, alt: "m ∡ 8 equals m ∡ 1 equals 140 degrees")[$m ∡ 8 = m ∡ 1 = 140^(∘)$] by alternate exterior angles.


        #math.equation(block: false, alt: "m ∡ 5 equals m ∡ 8 equals 140 degrees")[$m ∡ 5 = m ∡ 8 = 140^(∘)$] by vertical angles.
]
]


    
    #examplebox("Example 7")[Measuring Angles Formed by a Transversal][
In  given that angle 2 measures #math.equation(block: false, alt: "23 degrees ,")[$23^(∘) ,$] find the measure of the remaining angles and state the reason for your solution.


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  content((190.00pt, 46.00pt), anchor: "center", text(size: 9pt)[l₂])
})]),
  alt: "Two parallel lines, l subscript 1 and l subscript 2 are intersected by a transversal. The transversal makes four angles numbered 1, 23 degrees, 3, and 4 with the line, l subscript 1. The transversal makes four angles numbered 5, 6, 7, and 8 with the line, l subscript 2. 1, 23 degrees, 7, and 8 are exterior angles. 3, 4, 5, and 6 are interior angles. The corresponding angles, 1 and 5 are highlighted.", caption: none,
)


      
      #solutionbox[
#math.equation(block: false, alt: "m ∡ 2 equals m ∡ 3 equals 23 degrees")[$m ∡ 2 = m ∡ 3 = 23^(∘)$] by vertical angles, because #math.equation(block: false, alt: "∡ 2")[$∡ 2$] and #math.equation(block: false, alt: "∡ 3")[$∡ 3$] are the opposite angles formed by two intersecting lines.


        #math.equation(block: false, alt: "m ∡ 1 equals 157 degrees")[$m ∡ 1 = 157^(∘)$] by supplementary angles to #math.equation(block: false, alt: "m ∡ 2")[$m ∡ 2$] or #math.equation(block: false, alt: "m ∡ 3.")[$m ∡ 3.$] We see that #math.equation(block: false, alt: "∡ 1")[$∡ 1$] and #math.equation(block: false, alt: "∡ 2")[$∡ 2$] form a straight angle as does #math.equation(block: false, alt: "∡ 1")[$∡ 1$] and #math.equation(block: false, alt: "∡ 3.")[$∡ 3.$] A straight angle measures #math.equation(block: false, alt: "180 degrees ,")[$180^(∘) ,$] so #math.equation(block: false, alt: "180 degrees minus 23 degrees equals 157 degrees .")[$180^(∘) − 23^(∘) = 157^(∘) .$]


        #math.equation(block: false, alt: "m ∡ 4 equals m ∡ 1 equals 157 degrees")[$m ∡ 4 = m ∡ 1 = 157^(∘)$] by vertical angles, because #math.equation(block: false, alt: "∡ 4")[$∡ 4$] and #math.equation(block: false, alt: "∡ 1")[$∡ 1$] are the two opposite angles formed by two intersecting lines.


        #math.equation(block: false, alt: "m ∡ 5 equals m ∡ 1 equals 157 degrees")[$m ∡ 5 = m ∡ 1 = 157^(∘)$] by corresponding angles because they are the same angle formed by the transversal crossing two parallel lines, one exterior and one interior.


        #math.equation(block: false, alt: "m ∡ 8 equals m ∡ 5 equals 157 degrees")[$m ∡ 8 = m ∡ 5 = 157^(∘)$] by vertical angles because #math.equation(block: false, alt: "∡ 8")[$∡ 8$] and #math.equation(block: false, alt: "∡ 5")[$∡ 5$] are the two opposite angles formed by two intersecting lines.


        #math.equation(block: false, alt: "m ∡ 7 equals m ∡ 2 equals 23 degrees")[$m ∡ 7 = m ∡ 2 = 23^(∘)$] by alternate exterior angles because, like vertical angles, these angles are the opposite angles formed by the transversal intersecting two parallel lines.


        #math.equation(block: false, alt: "m ∡ 6 equals m ∡ 7 equals 23 degrees")[$m ∡ 6 = m ∡ 7 = 23^(∘)$] by vertical angles because these are the opposite angles formed by two intersecting lines.
]
]


    
    #examplebox("Example 8")[Finding Missing Angles][
Find the measures of the angles 1, 2, 4, 11, 12, and 14 in  and the reason for your answer given that #math.equation(block: false, alt: "l sub 1")[$l_(1)$] and #math.equation(block: false, alt: "l sub 2")[$l_(2)$] are parallel.


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  content((118.66pt, 30.12pt), text(size: 9pt)[6])
})]),
  alt: "Two parallel lines, l subscript 1 and l subscript 2 are intersected by two transversals. The first transversal makes four angles numbered 62 degrees, 9, 7, and 8 with the line, l subscript 2. The second transversal makes four angles numbered 14, 62 degrees, 11, and 12 with the line, l subscript 2. The two transversals intersect at a point on the line, l subscript 1. Six angles are formed around the intersection point. The angles are labeled 1, 2, 62 degrees, 4, 5, and 6.", caption: none,
)


      
      #solutionbox[
#math.equation(block: false, alt: "m ∡ 12 equals 118 degrees")[$m ∡ 12 = 118^(∘)$], supplementary angles


        #math.equation(block: false, alt: "m ∡ 14 equals 118 degrees")[$m ∡ 14 = 118^(∘)$], vertical angles


        #math.equation(block: false, alt: "m ∡ 11 equals 62 degrees")[$m ∡ 11 = 62^(∘)$], vertical angles


        #math.equation(block: false, alt: "m ∡ 4 equals 62 degrees")[$m ∡ 4 = 62^(∘)$], corresponding angles


        #math.equation(block: false, alt: "m ∡ 1 equals 62 degrees")[$m ∡ 1 = 62^(∘)$], vertical angles


        #math.equation(block: false, alt: "m ∡ 2 equals 56 degrees")[$m ∡ 2 = 56^(∘)$], supplementary angles
]
]


    
    #notebox("Who Knew?", rgb("#183B6F"), rgb("#183B6F"), rgb("#EFF1F5"))[
#emph[The Number 360]


      Did you ever wonder why there are #math.equation(block: false, alt: "360 degrees")[$360^(∘)$] in a circle? Why not #math.equation(block: false, alt: "100 degrees")[$100^(∘)$] or #math.equation(block: false, alt: "500 degrees ?")[$500^(∘) ?$] The number 360 was chosen by Babylonian astronomers before the ancient Greeks as the number to represent how many degrees in one complete rotation around a circle. It is said that they chose 360 for a couple of reasons: It is close to the number of days in a year, and 360 is divisible by 2, 3, 4, 5, 6, 8, 9, 10, …
]


  
  
  
  
    
=== Key Terms


    - vertex
- right angle
- acute angle
- obtuse angle
- straight angle
- complementary
- supplementary


  
  
    
=== Key Concepts


    - Angles are classified as acute if they measure less than #math.equation(block: false, alt: "90 degrees ,")[$90^(∘) ,$] obtuse if they measure greater than #math.equation(block: false, alt: "90 degrees")[$90^(∘)$] and less than #math.equation(block: false, alt: "180 degrees ,")[$180^(∘) ,$] right if they measure exactly #math.equation(block: false, alt: "90 degrees ,")[$90^(∘) ,$] and straight if they measure exactly #math.equation(block: false, alt: "180 degrees .")[$180^(∘) .$]
- If the sum of angles equals #math.equation(block: false, alt: "90 degrees")[$90^(∘)$], they are complimentary angles. If the sum of angles equals #math.equation(block: false, alt: "180 degrees")[$180^(∘)$], they are supplementary.
- A transversal crossing two parallel lines form a series of equal angles: alternate interior angles, alternate exterior angles, vertical angles, and corresponding angles


  
  
    
=== Formula


    To translate an angle measured in degrees to radians, multiply by #math.equation(block: false, alt: "the fraction π over 180 .")[$frac(π, 180) "."$]


    To translate an angle measured in radians to degrees, multiply by #math.equation(block: false, alt: "the fraction 180 over π .")[$frac(180, π) .$]