1.5 Exploring Angle Pairs

SOL: G4Objectives: The Student Will …• Identify and use special pairs of angles.• Use special angle pairs to determine angle measure.

Adjacent Angles

Have a common vertex Have a common side, but no common

interior points.

Two coplanar angles

Examples:∡ABC and ∡CBD

AB

D

C

Nonexamples:∡ABC and ∡ABD ∡ABC and ∡BCD

AC

DB

AC

DB

AB

CD

Are they Adjacent or Not???

ADB, BDC

A

B

CD

WVX, XVZ

OKN, MJL

W

X

YZ

V

LM

NO

K

J

Vertical Angles Formed by two intersecting lines Think of a bow tie For every set of intersecting lines there are two

sets of congruent angles

Are two nonadjacent angles

Examples: AEB and CED, AED and BEC∡ ∡ ∡ ∡

A B

CED

Are they Vertical or Not???

EFG, GFH

E F

G

H I

J

IHJ, EHJYVZ, WVX

W

X

Y

Z

V

YVZ, ZVW

XVY, WVZ

ZVW, WVX

Complementary Angles Two angles whose measures have a sum of 90 Examples:

1 and 2 are complementary

PQR and XYZ are complementary

12

P

R

Q Y

ZX

50°

40°

Example of Complimentary Angles

60

75

15

?

37

53

Supplementary Angles Two angles whose measures have a sum of 180 Example:

EFH and HFG are supplementary

M and N are supplementary

E F G

H

80°100°N

M

Examples of Supplementary Angles

13545

50130

Linear Pair Is a pair of adjacent angles

Common

Side

Noncommon Sides

Are opposite

rays

“stra

ight line”

Whose noncommon sides are opposite rays The angles of a linear pair forms a straight line

Example: ∡BED and ∡BEC

C

B

E

D

Are they a Linear Pair or Not???

EFG, GFH

E F

G

H I

J

IHJ, EHJYXZ, WXZ

YXW, WXZ

EFG, IHJ

W XY

Z

Example 1:Refer to the figure below. Name an angle pair that

satisfies each condition.

a.) two angles that form a linear pair.

b.) two acute vertical angles.

(2x + 24)° (4x + 36)°

Example 2: ∡KPL and ∡JPL are a linear pair, m∡KPL = 2x + 24, m 4∡ x + 36. What are the measures of ∡KPL and ∡JPL?

Since ∡KPL and ∡JPL are a linear pair, then we know their sum is 180°m∡KPL + m∡JPL = 180°

(2x + 24) + (4x + 36) = 180°

6x + 60 = 180°

- 60 - 60

6x = 120°

6x = 120°

6 6x = 20°

m∡KPL = 2(20) + 24 = 64°

m∡JPL = 4(20) + 36 = 116°

Angle Bisector A ray that divides an angle into two

congruent angles

Example: If PQ is the angle bisector of

RPS, then RPQ QPS

R Q

SP

Examples 3:

Angle Bisector

A

B

C

D

If mADB = 35,

Y

X

W

Z

If mYZX = 20,

then mBDC = ___then mWZX = ___

then mADC = ___ then mWZY = ___

35

70

2040

Example 4:If BX bisects ABC, find x and mABX

and mCBX.

A X

C

B

3x + 5

2x + 30

Bisector cuts and angle into two equal parts. Then m∡ABX = m∡CBX

m∡ABX = m∡CBX3x + 5 = 2x + 30

-2x -2x

x + 5 = 30- 5 -5

x = 25

m∡ABX = 3(25) + 5 = 80°

m∡CBX = 2(25) + 30 = 80°