1.6 describing pairs of angles - mesa public schools · 1.6 day 1 warm-up solve. 1. 4x − 0 ......

Geometry1.6 Describing Pairs of Angles

August 26, 20161.6 Describing Pairs of Angles

1.6 Day 1 Warm-up

Solve.

1. 4x − 0 = 12 2. 7 = −11c − 4

3. 11 = −19x − 8 4. 7 = 5n + 5 − 4n

5. 3x + 2 + 8 = 2x − 5 6. x + 5 + 6x + 17 = x − 2

Essential Question

What angle relationships occur when two

lines intersect?

August 26, 20161.4 Perimeter and Area in the Coordinate Plane

What You Will Learn

• Identify complementary and supplementary angles.

• Identify linear pairs, vertical angles, and adjacent angles.

• Solve problems with angle relationship properties.


Adjacent Angles


A

B

CO

Adjacent angles have the

same vertex, O, and one side

in common, OB. They share

no interior points.

There are THREE angles:

AOB or BOA

BOC or COB

AOC or COA

You cannot use the label

O, since it would be

unclear which angle that is.

RST and VST are NOT adjacent angles.


R

S T

V

Why not?

They overlap.

Linear Pair


1 2

Two adjacent angles are a linear pair if their noncommon

sides are opposite rays.

Common Side

Noncommon sides

1 & 2 are

a linear pair.

Linear Pair Property


The sum of the angles of a linear pair is 180°.

70° ?110°

Complementary Angles


Two angles are complementary if their sum is 90°.

65°

25°

These angles are

complementary

and adjacent.

Complementary Angles


Two angles are complementary if their sum is 90°.

These angles are

complementary

and

nonadjacent.

30°

60°

Supplementary Angles


Angles are supplementary if their sum is 180°.

70° 110°

These angles are

supplementary and

adjacent and a linear pair.

Supplementary Angles


Angles are supplementary if their sum is 180°.

The angles are

supplementary

and nonadjacent.

80° 100°

Example 1


In the figure, name a pair of complementary angles, a

pair of supplementary angles, and a pair of adjacent

angles.

Example 2


Vertical Angles


1 23

4

Two angles are vertical

angles if their sides form

two pairs of opposite rays.

1 & 2 are

vertical angles.

3 & 4 are

vertical angles.

Vertical Angles Property


Vertical Angles are congruent.

60°?60°

Example 3


1

2 3

45

a. Are 1 and 2 a linear pair?

Yes

b. Are 4 and 5 a linear pair?

No

c. Are 3 and 5 vertical angles?

No

d. Are 1 and 3 vertical angles?

Yes

Example 4


50° 12

3

Find the measure of the three angles.

These angles are vertical angles.

Vertical angles are congruent.

50°

These angles form a

linear pair. The sum is

180°.

130°

These are vertical

angles, and

congruent.

130°

Example 5


A B

DC

E

(4x + 30)°

(6x – 10)°

Solve for x, then

find the measure of

each angle.

AEB and BEC

form a linear pair.

What do we know about the sum of the angles of

a linear pair? The sum is 180°.

Example 5


A B

DC

E

(4x + 30)°

(6x – 10)°

Linear pair AEB and

BEC means:

(4x + 30) + (6x – 10) = 180

10x + 20 = 180

10x = 160

x = 16

Then AEB = 4(16) + 30 = 94

and BEC = 6(16) – 10 = 86

94°

86°

94°

86°

Your Turn


Work through these two problems.

145°1

23

1. Find the measure

of 1, 2, 3.

(2x – 4)°(5x + 30)°

A B

C

2. Find the measure

of ABC.

Your Turn Solutions


145°1

23

(2x – 4)°(5x + 30)°

A B

C

145°

35°

35°

180°

5x + 30 + 2x – 4 = 180

7x + 26 = 180

7x = 154

x = 22

mABC = 5(22) + 30

= 140°

Example 6


A

B

C

D

6x°

(3x + 45)°

Solve for x, then find the angle measures.

Solution:

AEB and DEA are a

linear pair. The sum of

the angles in a linear pair

is 180°.

6x + (3x + 45) = 180

9x = 135

x = 15

E

6(15) = 90°

3(15) + 45 = 90°

Example 7


(5y – 50)°

(4y – 10)°

1

Solve for y, then find m1.

Vertical angles are

congruent, so:

5y – 50 = 4y – 10

y = 40

5(40) – 50 = 150°

150° 1 forms a linear pair with

either of the 150° angles, so

1 is 30°.

30°

Example 8


(4x + 5)°

(3x + 8)°

Find the measure of each angle.

This is a right angle, the

angles are complementary.

Their sum is 90°.

4x + 5 + 3x + 8 = 90

7x + 13 = 90

7x = 77

x = 11

4(11) + 5 = 49°

49°

3(11) + 8 = 41°

41°

Example 9


(3x + 8)° (5x – 20)°

Find the value of each variable and the measure of each

labeled angle.

3x + 8 =5x – 20

-2x = -28

x = 14 3(14) + 8 = 50°

50°50°

130°

1. Solve for x.


(6x + 10)(4x + 40)

6 10 4 40

2 30

15

x x

x

x

2. Solve for x.


(5x + 5)(12x – 12)

(12 12) (5 5) 180

17 7 180

17 187

11

x x

x

x

x

3. Solve for x.


(7x + 2)

( 8) (7 2) 90

8 10 90

8 80

10

x x

x

x

x

4. Solve for x & y.


(7x + 4)

(13x + 16)

(9y + 3)

(5y 5)

(7 4) (13 16) 180

20 20 180

20 160

8

(9 3) (5 5) 180

14 2 180

14 182

13

x

y y

x

x

x

y

y

y

x

5. Solve for x.


A is supplementary to B.

mA = (2x + 10)

mB = (3x 5)

2x + 10 + 3x 5 = 180

5x + 5 = 180

5x = 175

x = 35

Quick Review

Two angles are complementary if their sum is 90.


What do you know about supplementary angles?

Two angles are supplementary if their sum is 180.

What do you know about complementary angles?

Quick Review


1 2

4

3

Which angles are Vertical Angles and what do you

know about them?

1 2 & 3 4

Quick Review


1 2

4

3

Which angles are linear with ∠4 and what do

we know about them?

m1 + m4 = 180

m4 + m2 = 180

Essential Question

When two lines intersect, how do you

know if two angles are congruent or

supplementary and how do you use this

information to find angle measures?

August 26, 20161.4 Perimeter and Area in the Coordinate Plane

Assignment


1.6 describing pairs of angles - mesa public schools · 1.6 day 1 warm-up solve. 1. 4x − 0 ......

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