Vertical Angles and Linear Pairs

Previously, you learned that two angles are adjacent if they share a common vertex and side but have no common interior points. In this lesson, you will study other relationships between pairs of angles.

1 and 3 are vertical angles.

2 and 4 are vertical angles.

14

3

2

Two angles are vertical angles if their sides form two pairs of opposite rays.

Complementary and Supplementary Angles

Two angles are complementary angles if the sum of their measurements is 90˚. Each angle is the complement of the other. Complementary angles can be adjacent or nonadjacent.

12 3

4

complementary adjacent

complementary nonadjacent

Complementary and Supplementary Angles

5 67 8

supplementary nonadjacent

supplementary adjacent

Two angles are supplementary angles if the sum of their measurements is 180˚. Each angle is the supplement of the other. Supplementary angles can be adjacent or nonadjacent.

Complimentary and Supplementary Angles

Two angles that are complimentary:

Two angles that are supplementary

40

50

130

50

Finding Measures of Complements and Supplements

Find the angle measure.

SOLUTION

mC = 90˚ – m A

= 90˚ – 47˚

= 43˚

Given that A is a complement of C and m A = 47˚, find mC.

Finding Measures of Complements and Supplements

Find the angle measure.

SOLUTION

mC = 90˚ – m A

= 90˚ – 47˚

= 43˚

Given that A is a complement of C and m A = 47˚, find mC.

mP = 180˚ – mR

= 180 ˚ – 36˚

= 144˚

Given that P is a supplement of R and mR = 36˚, find mP.

Classwork due at end

Page 51-52 1-20

Warm up

On little piece of paper, number 1-5 and write always, sometimes or never.

1.Vertical angles ________ have a common vertex.

2.Two right angles are ______ complementary.3.Right angles are _______ vertical angles.4.Angles A, B and C are ______ complementary.5.Vertical angles ______ have a common

supplement.

Putting Vertical Angles to Use

Using Algebra to find Vertical Angles

Solve for x and y. Then find the angle measure.

( x + 15)˚

( 3x + 5)˚( y + 20)˚

( 4y – 15)˚

D•

•C • B

A •E

SOLUTIONUse the fact that the sum of the measures of angles that form a linear pair is 180˚.

m AED + m DEB = 180°

( 3x + 5)˚ + ( x + 15)˚ = 180°

4x + 20 = 180

4x = 160

x = 40

m AEC + mCEB = 180°

( y + 20)˚ + ( 4y – 15)˚ = 180°

5y + 5 = 180

5y = 175

y = 35

Use substitution to find the angle measures (x = 40, y = 35).

m AED = ( 3 x + 15)˚ = (3 • 40 + 5)˚

m DEB = ( x + 15)˚ = (40 + 15)˚

m AEC = ( y + 20)˚ = (35 + 20)˚

m CEB = ( 4 y – 15)˚ = (4 • 35 – 15)˚

= 125˚

= 55˚

= 55˚

= 125˚

So, the angle measures are 125˚, 55˚, 55˚, and 125˚. Because the vertical angles are congruent, the result is reasonable.

W and Z are complementary. The measure of Z is 5 times the measure of W. Find m W

SOLUTION

Because the angles are complementary,

But m Z = 5( m W ),

Because 6(m W) = 90˚,

so m W + 5( m W) = 90˚.

you know that m W = 15˚.

m W + m Z = 90˚.

Right Now!!!!!!

Complete pages 52-53 2-20 even (skip 6), 24-26 evens, 30Answers 2) 17.5, 107.54) 90-2y, 180-2y8) AFE & EFD; AFE & BFC10) BFE & EFD; CFA & AFE; DFC & CFB12) BFE & CFD14) 15516) 12018) 8520) X = 1024) X = 65, A = 130, B = 5026) Y = 17, C = 56, D = 3430) 180 – X = 2X + 12, X = 56, 124

Definition:

Perpendicular Lines

Perpendicular lines are two lines that intersect to form right angles.

90

Perpendicular lines are also intersecting lines because they cross each other.

Perpendicular lines are a special kind of intersecting lines because they always form “perfect” right angles.

90

Theorems

Theorem 2-4:If two lines are perpendicular, then they form congruent adjacent angles.

Theorem 2-5:If two lines form congruent adjacent angles, then the lines are perpendicular.

Theorem 2-6:If the exterior sides of two adjacent acute angles are perpendicular, then the angles are complementary.

Example

Name the definition of state the theorem that justifies the statement about the diagram.

1. If 6 is right angle, then RS ⊥ TV.2. If RS ⊥ TV, then 5, 6, 7, and 8 are right angles.3. If RS ⊥ TV, then 8 ≃ 7.4. If RS ⊥ TV, then m 6 = 90.5. If 5 ≃ 6, then RS ⊥ TV.6. If m 5 = 90, then RS ⊥ TV.

1. Def of ⊥ lines 6. Def of ⊥ lines2. Def of ⊥ lines3. Thm. 2-44. Def of ⊥ lines5. Thm. 2-5

A proof!

Statement Reasons

1. 1 ≃ 2, or m1 = m2 1.

2. m1 + m2 = 180 2.

3. m2 + m2 = 180, or 2m2 = 180

3.

4. m2 = 90 4.

5. _____ 5.

Classwork due by end of class

Page 58 – 59 3-8 13

Closure

Complete with always, sometimes or never.

1. Perpendicular lines _______ line in the same plane.

2. Two lines are perpendicular if and only if they ______ form congruent adjacent angles.

3. Perpendicular lines ______ form 60 degree angles.

4. If a pair of vertical angles are supplementary, the lines forming the angles are _____ perpendicular.

Homework

Page 52-54 1-21 odds (skip 5) 25, 27, 29, 32, 33

Page 58-59 9-12 14-25