Identifying Pairs of Angles
A pair of angles can sometimes be classified by their combined measure.
Complementary Angles โ Two angles are complementary if the sum of their measures is 90ยฐ.
๐โ ๐ด๐ต๐ถ + ๐โ ๐ถ๐ต๐ท = 90ยฐ, so โ ๐ด๐ต๐ถ is complementary to โ ๐ถ๐ต๐ท.
Supplementary Angles - Two angles are supplementary if the sum of their measures is 180ยฐ.
๐โ ๐๐๐ + ๐โ ๐ ๐๐ = 180ยฐ, so โ ๐๐๐ is supplementary to โ ๐ ๐๐.
a. Find the angles complementary to โ ๐พ๐ฟ๐ if ๐โ ๐พ๐ฟ๐ = 90ยฐ.
Solution
From the diagram, ๐โ ๐ฝ๐ฟ๐พ + ๐โ ๐พ๐ฟ๐ = 90ยฐ and ๐โ ๐พ๐ฟ๐ + ๐โ ๐๐ฟ๐ = 90ยฐ. So โ ๐ฝ๐ฟ๐พ and โ ๐๐ฟ๐ are complementary to โ ๐พ๐ฟ๐.
Find the angles supplementary to โ ๐ท๐บ๐น.
Solution
From the diagram, ๐โ ๐ธ๐บ๐ท + ๐โ ๐ท๐บ๐น = 180ยฐ and ๐โ ๐ท๐บ๐น + ๐โ ๐น๐บ๐ถ = 180ยฐ. So โ ๐ธ๐บ๐ท and โ ๐น๐บ๐ถ are supplementary to โ ๐ท๐บ๐น.
Theorem 6-1: Congruent Complements Theorem โ If two angles are complementary to the same angle or to congruent angles, then they are congruent.
Theorem 6-2: Congruent Supplements Theorem - If two angles are supplementary to the same angle or to congruent angles, then they are congruent.
Find the measures of the angles labeled x and y.
Solution
To find x, notice that โ ๐ท๐ต๐น and โ ๐น๐ต๐ธ are complementary.
๐โ ๐ท๐ต๐น + ๐โ ๐น๐ต๐ธ = 90ยฐ Definition of comp. angles.
55ยฐ + ๐ฅ = 90ยฐ Substitute.
55ยฐ + ๐ฅ โ 55ยฐ = 90ยฐ โ 55ยฐ Subtract 55ยฐ from each side.
๐ฅ = 35ยฐ Simplify
Find the measures of the angles labeled x and y.
Solution
To find y, notice that โ ๐ด๐ต๐ท and โ ๐ท๐ต๐ถ are supplementary.
๐โ ๐ด๐ต๐ท + ๐โ ๐ท๐ต๐ถ = 180ยฐ Definition of supp. angles.
๐ฆ + 55ยฐ + 35ยฐ + 40ยฐ = 180ยฐ Substitute.
๐ฆ + 130ยฐ = 180ยฐ Simplify.
๐ฆ + 130ยฐ โ 130 = 180ยฐ โ 130ยฐ Sub. 130ยฐ from each side.
๐ฅ = 50ยฐ Simplify
Adjacent Angles - Two angles in the same plane that share a vertex and a side, but share no interior points. In the diagram, โ ๐๐๐ฟ is adjacent to โ ๐ฟ๐๐ and โ ๐ ๐๐ is adjacent to โ ๐๐๐ฟ.
Linear Pair - Adjacent angles whose non-common sides are opposite rays. In the diagram โ ๐ ๐๐ and โ ๐๐๐ are a linear pair. Linear pairs are also supplementary because their measures add up to 180ยฐ.
Theorem 6-3: Linear Pair Theorem โ If two angle form a linear pair, then they are supplementary.
Identify two sets of adjacent angles and one linear pair.
Solution
There are many adjacent angles in the diagram. Two possible sets are โ ๐ด๐น๐ต and โ ๐ต๐น๐ถ, and โ ๐ด๐น๐ถ and โ ๐ถ๐น๐ธ.
There are also several linear pairs. One is โ ๐ด๐น๐ท and โ ๐ท๐น๐ธ.
Vertical Angles โ Nonadjacent angles formed by two intersecting lines.
Theorem 6-4: Vertical Angle Theorem โ If two angles are vertical angles, then they are congruent.
Determine the values of x and y.
Solution
Since โ 1 and โ 2 are vertical angles, they are congruent. The same is true of โ 3 and โ 4. Therefore, ๐โ 1 = ๐โ 2 and ๐โ 3 = ๐โ 4.
๐โ 1 = ๐โ 2 2๐ฅ โ 5 = ๐ฅ + 30 2๐ฅ โ 5 + 5 = ๐ฅ + 30 + 5
2๐ฅ โ ๐ฅ = ๐ฅ + 35 โ ๐ฅ ๐ฅ = 35
Determine the values of x and y. Solution Since โ 1 and โ 2 are vertical angles, they are congruent. The same is true of โ 3 and โ 4. Therefore, ๐โ 1 = ๐โ 2 and ๐โ 3 = ๐โ 4. ๐โ 3 = ๐โ 4 ๐ฆ + 17 = 2๐ฆ โ 81 ๐ฆ + 17 โ 17 = 2๐ฆ โ 81 โ 17 ๐ฆ = 2๐ฆ โ 89 ๐ฆ โ 2๐ฆ = 2๐ฆ โ 98 โ 2๐ฆ โ๐ฆ = โ98 ๐ฆ = 89
The diagram shows the part of a bridge where it contacts a vertical cliff, so that the bridge and the cliff are perpendicular. The angle between the surface of the road and the line extended from the bridgeโs support measures 50ยฐ. It is important that the bridgeโs support be set at the correct angle to hold the weight of the bridge. What is the angle x that the support makes with the cliff?
Solution
The angle that measures 50ยฐ and the angle labeled y are vertical angles. The angles labeled x and y are complementary angles. ๐ฅ + ๐ฆ = 90ยฐ ๐ฅ + 50ยฐ = 90ยฐ ๐ฅ + 50ยฐ โ 50ยฐ = 90ยฐ โ 50ยฐ ๐ฅ = 40ยฐ
Find the value of x.
๐ฅ = 33
Determine the value of x.
๐ฅ = 8
Page 37
Lesson Practice a-f (Ask Mr. Heintz)
Page 38
Practice 1-30 (Do the starred ones first)