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Geometry Lesson 1.6
Angle Pair Relationships
Warm-Up: Angle Types (review)

Name the type of angle and state how
many degrees it can have
Right angle
(exactly 90°)
Obtuse angle
(> 90° and < 180°)
Acute angle
< 90°
Straight angle
(exactly 180°)
1. Vertical Angles
Vertical Angles: Two angles whose
sides form two pairs of opposite rays
 In this context, “vertical” means
“shared vertex”, not “straight up”

4
1
3
2
1 and 3 are vertical angles
2 and 4 are vertical angles
2. Linear Pairs

Linear Pair: Two adjacent angles
whose non-common sides are
opposite rays
1
2
Adjacent
angles have a
common side
Non-common sides are opposite rays
1 and 2 are a linear pair
Example 1: Identifying Angle Pairs

Are 1 & 2 adjacent?
Yes: Common side

Are 1 & 2 a linear pair?
Yes: Adjacent & opposite rays

Are 3 & 4 a linear pair?
No: Adjacent, but not opposite rays

Are 2 & 5 vertical angles?
Yes: Two pairs of opposite rays

Are 1 & 4 vertical angles?
No: Sides are NOT opposite rays

Are 3 & 5 vertical angles?
No: Sides are NOT opposite rays
Practice 1: Identify Angle Pairs

Answer the questions for each figure
(a)
(b)
No
No
Yes
No
Yes
No
Yes
No
3. Properties of Angle Pairs

The sum of the angle measures in a
linear pair is always 180°
1

m1 + m2 = 180°
2
Vertical Angles are always congruent
(equal measures)
4
1
3
2
m1 = m3
m2 = m4
Simulations

Vertical angles animation

Linear pair animation
Example 2a: Finding Angle Measures
129°
Linear pair: 51° + m7 = 180°
Vertical angles are congruent
Linear pair: 136° + m8 = 180°
Vertical angles are congruent
103°
44°
53°
Example 2b: Finding Angle Measures


This time, let’s use algebra…
Find the value of x and use it
to find the angle measures
115°
FHI  GHJ (vertical s)
115°
mFHI  mGHJ
(7x – 25)° = (5x + 15)°
mFHI = 7(20) – 25
2x = 40  x = 20
mFHI = 115°
Now, substitute x…
mGHJ = 5(20) + 15
mGHJ = 115°
Practice 2: Finding Angle Measures
Find x or y and then evaluate the angle
measures
(b)
(a)

4. Complementary Angles

Complementary Angles: Two angles
whose measures total 90°
1
2
Adjacent
or
1
2
Non-adjacent
1 and 2 are complementary
m1 + m2 = 90°
5. Supplementary Angles

Supplementary Angles: Two angles
whose measures total 180°
5
6
Adjacent
or
5
6
Non-adjacent
5 and 6 are supplementary
m1 + m2 = 180°
Example 3a: Complements & Supplements
E is a complement of F
 If mE = 68°, find mF
E F
 mE + mF = 90°
 68° + mF = 90°  mF = 22°
 G is a supplement of H
 If mG = 152°, find mH
G H
 mG + mH = 180°
 152° + mH = 180°  mH = 28°

Example 3b: Comp & Supp w/Algebra





A is supplementary to B
A is complementary to C
A B
mA = x°; mB = (x + 40)°
mC = (x – 50)°
A C
Find all angle measures
mA + mB = 180°
2x + 40 = 180  x = 70
mA = 70°
mB = (70 + 40)° = 110°
mC = (70 – 50)° = 20°
Practice 3: Comp & Supp s
(a) A is a complement of B and mA = 81°
Find mB
(b) C is a supplement of D and mC = 27°
Find mD
(c) Repeat example 3b with the following:
mA = x°; mB = (2x)°; and mC = (x – 30)°
Closure: Angle Pairs
Angle measures in a linear pair add up
to ________°
 Angle measures in vertical angles are
_____________
 Complementary angle measures add
up to _______°
 Supplementary angle measures add
up to _______°

For some cool computer animations, go to
http://www.mathopenref.com/tocs/anglestoc.html
Complementary Angles
You’re sooo acute!!
Homework
1.6 w/s