Identifying Pairs of Angles
Identifying Pairs of Angles
A pair of angles can sometimes be classified by their combined measure.
Complementary Angles β Two angles are complementary if the sum of their measures is 90Β°.
πβ π΄π΅πΆ + πβ πΆπ΅π· = 90Β°, so β π΄π΅πΆ is complementary to β πΆπ΅π·.
Supplementary Angles - Two angles are supplementary if the sum of their measures is 180Β°.
πβ πππ + πβ π ππ = 180Β°, so β πππ is supplementary to β π ππ.
a. Find the angles complementary to β πΎπΏπ if πβ πΎπΏπ = 90Β°.
Solution
From the diagram, πβ π½πΏπΎ + πβ πΎπΏπ = 90Β° and πβ πΎπΏπ + πβ ππΏπ = 90Β°. So β π½πΏπΎ and β ππΏπ are complementary to β πΎπΏπ.
Find the angles supplementary to β π·πΊπΉ.
Solution
From the diagram, πβ πΈπΊπ· + πβ π·πΊπΉ = 180Β° and πβ π·πΊπΉ + πβ πΉπΊπΆ = 180Β°. So β πΈπΊπ· and β πΉπΊπΆ are supplementary to β π·πΊπΉ.
Theorem 6-1: Congruent Complements Theorem β 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 β π·π΅πΉ and β πΉπ΅πΈ are complementary.
πβ π·π΅πΉ + πβ πΉπ΅πΈ = 90Β° Definition of comp. angles.
55Β° + π₯ = 90Β° Substitute.
55Β° + π₯ β 55Β° = 90Β° β 55Β° Subtract 55Β° from each side.
π₯ = 35Β° Simplify
Find the measures of the angles labeled x and y.
Solution
To find y, notice that β π΄π΅π· and β π·π΅πΆ are supplementary.
πβ π΄π΅π· + πβ π·π΅πΆ = 180Β° Definition of supp. angles.
π¦ + 55Β° + 35Β° + 40Β° = 180Β° Substitute.
π¦ + 130Β° = 180Β° Simplify.
π¦ + 130Β° β 130 = 180Β° β 130Β° Sub. 130Β° from each side.
π₯ = 50Β° Simplify
Adjacent Angles - Two angles in the same plane that share a vertex and a side, but share no interior points. In the diagram, β πππΏ is adjacent to β πΏππ and β π ππ is adjacent to β πππΏ.
Linear Pair - Adjacent angles whose non-common sides are opposite rays. In the diagram β π ππ and β πππ are a linear pair. Linear pairs are also supplementary because their measures add up to 180Β°.
Theorem 6-3: Linear Pair Theorem β 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 β π΄πΉπ΅ and β π΅πΉπΆ, and β π΄πΉπΆ and β πΆπΉπΈ.
There are also several linear pairs. One is β π΄πΉπ· and β π·πΉπΈ.
Vertical Angles β Nonadjacent angles formed by two intersecting lines.
Theorem 6-4: Vertical Angle Theorem β If two angles are vertical angles, then they are congruent.
Determine the values of x and y.
Solution
Since β 1 and β 2 are vertical angles, they are congruent. The same is true of β 3 and β 4. Therefore, πβ 1 = πβ 2 and πβ 3 = πβ 4.
πβ 1 = πβ 2 2π₯ β 5 = π₯ + 30 2π₯ β 5 + 5 = π₯ + 30 + 5
2π₯ β π₯ = π₯ + 35 β π₯ π₯ = 35
Determine the values of x and y. Solution Since β 1 and β 2 are vertical angles, they are congruent. The same is true of β 3 and β 4. Therefore, πβ 1 = πβ 2 and πβ 3 = πβ 4. πβ 3 = πβ 4 π¦ + 17 = 2π¦ β 81 π¦ + 17 β 17 = 2π¦ β 81 β 17 π¦ = 2π¦ β 89 π¦ β 2π¦ = 2π¦ β 98 β 2π¦ βπ¦ = β98 π¦ = 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 bridgeβs support measures 50Β°. It is important that the bridgeβs 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 50Β° and the angle labeled y are vertical angles. The angles labeled x and y are complementary angles. π₯ + π¦ = 90Β° π₯ + 50Β° = 90Β° π₯ + 50Β° β 50Β° = 90Β° β 50Β° π₯ = 40Β°
Find the value of x.
π₯ = 33
Determine the value of x.
π₯ = 8
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Lesson Practice a-f (Ask Mr. Heintz)
Page 38
Practice 1-30 (Do the starred ones first)