Angles of Triangles

Section 4.2

Lesson Quiz

Use the properties of similar figures to answer each question.

1. A rectangular house is 32 ft wide and 68 ft long. On a blueprint, the width is 8 in. Find the length on the blueprint.

2. Karen enlarged a 3 in. wide by 5 in. tall photo into a poster. If the poster is 2.25 ft wide, how tall is it?

3. Which rectangles are similar?

17 in.

3.75 ft

A and B are similar.

Bell Ringer

Objectives

Find angle measures in triangles.

Key Vocabulary

Corollary

Exterior angles

Interior angles

Theorems

4.1 Triangle Sum Theorem

Corollary to the Triangle Sum Theorem

4.2 Exterior Angle Theorem

Measures of Angles of a Triangle

The word “triangle” means “three angles”

When the sides of a triangles are extended,

however, other angles are formed

The original 3 angles of the triangle are the

interior angles

The angles that are adjacent to interior angles

are the exterior angles

Each vertex has a pair of exterior angles

Original Triangle Extend sides

Interior

Angle

Exterior

Angle

Exterior

Angle

Triangle Interior and Exterior Angles

A

B

C

Smiley faces are interior

angles and hearts

represent the exterior

angles

Each vertex has a pair

of congruent exterior

angles; however it is

common to show only

one exterior angle at

each vertex.

Triangle Interior and Exterior Angles

)))

A

B C

( D

E F

Interior Angles

Exterior Angles

(formed by extending the sides)

Triangle Sum Theorem

The Triangle Angle-Sum Theorem gives

the relationship among the interior angle

measures of any triangle.

Triangle Sum Theorem

If you tear off two corners of a triangle and

place them next to the third corner, the

three angles seem to form a straight line.

You can also show this in a drawing.

Draw a triangle and extend one side. Then

draw a line parallel to the extended side, as

shown.

The three angles in the triangle can be

arranged to form a straight line or 180°.

Two sides of the

triangle are

transversals to the

parallel lines.

Triangle Sum Theorem

Theorem 4.1 – Triangle Sum Theorem

The sum of the measures of the angles of a

triangle is 180°.

mX + mY + mZ = 180°

X

Y Z

Triangle Sum Theorem

Given mA = 43° and mB = 85°, find mC.

ANSWER C has a measure of 52°.

CHECK Check your solution by substituting 52° for mC. 43° +

85° + 52° = 180°

SOLUTION

mA + mB + mC = 180° Triangle Sum Theorem

43° + 85° + mC = 180° Substitute 43° for mA and

85° for mB.

128° + mC = 180° Simplify.

mC = 52° Simplify.

128° + mC – 128° = 180° – 128° Subtract 128° from each side.

Example 1

A. Find p in the acute triangle.

73° + 44° + p° = 180°

117 + p = 180

p = 63

–117 –117

Triangle Sum

Theorem

Subtract 117 from

both sides.

Example 2a

B. Find m in the obtuse triangle.

23° + 62° + m° = 180°

85 + m = 180

m = 95

–85 –85

Triangle Sum

Theorem

Subtract 85 from

both sides.

23

62

m

Example 2b

A. Find a in the acute triangle.

88° + 38° + a° = 180°

126 + a = 180

a = 54

–126 –126

88°

38°

a°

Triangle Sum

Theorem

Subtract 126

from both sides.

Your Turn:

B. Find c in the obtuse triangle.

24° + 38° + c° = 180°

62 + c = 180

c = 118

–62 –62 c°

24°

38° Triangle Sum

Theorem.

Subtract 62 from

both sides.

Your Turn:

2x° + 3x° + 5x° = 180°

10x = 180

x = 18

10 10

Find the angle measures in the scalene triangle.

Triangle Sum Theorem

Simplify.

Divide both sides by 10.

The angle labeled 2x° measures

2(18°) = 36°, the angle labeled

3x° measures 3(18°) = 54°, and

the angle labeled 5x° measures

5(18°) = 90°.

Example 3

3x° + 7x° + 10x° = 180°

20x = 180

x = 9

20 20

Find the angle measures in the scalene triangle.

Triangle Sum Theorem

Simplify.

Divide both sides by 20.

3x° 7x°

10x° The angle labeled 3x°

measures 3(9°) = 27°, the

angle labeled 7x°

measures 7(9°) = 63°, and

the angle labeled 10x°

measures 10(9°) = 90°.

Your Turn:

Find the missing angle measures.

Find first because the measure of two angles of the triangle are known.

Angle Sum Theorem

Simplify.

Subtract 117 from each side.

Example 4:

Answer:

Angle Sum Theorem

Simplify.

Subtract 142 from each side.

Example 4:

Find the missing angle measures.

Answer:

Your Turn:

Exterior Angles and Triangles

An exterior angle is formed by one side of a triangle and the extension of another side (i.e. 1 ).

The interior angles of the triangle not adjacent to a given exterior angle are called the remote interior angles (i.e. 2 and 3).

1 2

3 4

Investigating Exterior Angles of a

Triangles

B

A

A B

C

You can put the two torn angles

together to exactly cover one of the

exterior angles

Theorem 4.2 – Exterior Angle Theorem

The measure of an exterior angle of a

triangle is equal to the sum of the

measures of the two remote interior

angles.

m 1 = m 2 + m 3

1 2

3 4

ANSWER 1 has a measure of 130°.

SOLUTION

m1 = mA + mC Exterior Angle Theorem

Given mA = 58° and mC = 72°, find m1.

Substitute 58° for mA and

72° for mC. = 58° + 72°

Simplify. = 130°

Example 7

ANSWER 120°

ANSWER 155°

ANSWER 113°

Find m2. 1.

Find m3. 2.

Find m4. 3.

Your Turn:

Find the measure of each numbered angle in the figure.

Exterior Angle Theorem

Simplify.

Substitution

Subtract 70 from each side.

If 2 s form a linear pair, they are supplementary.

Example 8:

Exterior Angle Theorem

Subtract 64 from each side.

Substitution

Subtract 78 from each side.

If 2 s form a linear pair, they are supplementary. Substitution

Simplify.

Example 8: m∠1=70

m∠2=110

Subtract 143 from each side.

Angle Sum Theorem

Substitution

Simplify.

Answer:

Example 8: m∠1=70

m∠2=110

m∠3=46

m∠4=102

Find the measure of each numbered angle in the figure.

Answer:

Your Turn:

Joke Time

What's orange and sounds like a parrot?

A carrot!

What do you call cheese that doesn't belong to

you?

Nacho cheese.

Why do farts smell?

So the deaf can enjoy them too.

Assignment

Pg. 182-184: #1 – 13 odd, 19 – 29 odd