Angles of Triangles 3-4. EXAMPLE 1 Classify triangles by sides and by angles SOLUTION The triangle has a pair of congruent sides, so it is isosceles.

Angles of TrianglesAngles of TrianglesAngles of TrianglesAngles of Triangles

3-43-4

EXAMPLE 1 Classify triangles by sides and by angles

SOLUTION

The triangle has a pair of congruent sides, so it is isosceles. By measuring, the angles are 55° , 55° , and 70° . It is an acute isosceles triangle.

Support Beams

Classify the triangular shape of the support beams in the diagram by its sides and by measuring its angles.

EXAMPLE 2 Classify a triangle in a coordinate plane

SOLUTION

STEP 1 Use the distance formula to find the side lengths.

Classify PQO by its sides. Then determine if the triangle is a right triangle.

OP = y2 – y1( )2x2 – x1( )2 +

= 2 – 0( )2(– 1 ) 0( )2 +– = 5 2.2

OQ = y2 – y1( )2x2 – x1( )2 +

2= – 0( )6 0( )2 +– 3 = 45 6.7

EXAMPLE 2 Classify a triangle in a coordinate plane

PQ = y2 – y1( )2x2 – x1( )2 +

3 – 2( )26( )2 +–= (– 1 ) = 50 7.1

STEP 2 Check for right angles.

The slope of OP is 2 – 0 – 2 – 0

= – 2.

The slope of OQ is 3 – 0 6 – 0

=21 .

1The product of the slopes is – 2

2 = – 1,

so OP OQ and POQ is a right angle.

Therefore, PQO is a right scalene triangle.ANSWER

GUIDED PRACTICE for Examples 1 and 2

1. Draw an obtuse isosceles triangle and an acute scalene triangle.

obtuse isosceles triangle

B

A C

acute scalene triangleP

Q

R

GUIDED PRACTICE for Examples 1 and 2

2. Triangle ABC has the vertices A(0, 0), B(3, 3), and C(–3, 3). Classify it by its sides. Then determine if it is a right triangle.

SOLUTION

STEP 1 Use the distance formula to find the side lengths.

AB = y2 – y1( )2x2 – x1( )2 +

= 3 – 0( )2( 3 ) 0( )2 +–

BC = y2 – y1( )2x2 x1( )2 +

2= – 3( )–3

3( )2 +– 3

= 18 4.2

= 400 20

GUIDED PRACTICE for Examples 1 and 2

AC = y2 – y1( )2x2 – x1( )2 +

= 3 – 0( )2 0 )(–3( )2 +– = 18 4.2

STEP 2 Check for right angles.

The slope of AB is 3 – 0 3 – 0

= 1.

The product of the slopes is 1(– 1) = – 1,

so AB AC and BAC is a right angle.

The slope of AC is 3 – 0 – 3 – 0 = .– 1

Therefore, ABC is a right Isosceles triangle.

ANSWER

EXAMPLE 3 Find an angle measure

SOLUTION

STEP 1 Write and solve an equation to find the value of x.

Apply the Exterior Angle Theorem.(2x – 5)° =70° + x°

Solve for x.x = 75

STEP 2Substitute 75 for x in 2x – 5 to find m∠JKM.

2x – 5 = 2 75 – 5 = 145

ALGEBRA Find m∠JKM.

The measure of ∠JKM is 145°.ANSWER

EXAMPLE 4 Find angle measures from a verbal description

ARCHITECTURE

The tiled staircase shown forms a right triangle. The measure of one acute angle in the triangle is twice the measure of the other. Find the measure of each acute angle.

SOLUTION

First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

EXAMPLE 4 Find angle measures from a verbal description

Use the corollary to set up and solve an equation.

Corollary to the Triangle Sum Theoremx° + 2x° = 90°

Solve for x.x = 30

So, the measures of the acute angles are 30° and 2(30°) = 60° .

ANSWER

GUIDED PRACTICE for Examples 3 and 4

SOLUTION

STEP 1 Write and solve an equation to find the value of x.

Apply the Exterior Angle Theorem. (5x – 10)° = 40° + 3x°

Solve for x.2x =50

Find the measure of 1 in the diagram shown.3.

x=25

GUIDED PRACTICE for Examples 3 and 4

STEP 2 Substitute 25 for x in 5x – 10 to find 1.

5x – 10 = 5 25– 10 = 115

1 + (5x – 10)° = 180

1 + 115° = 180°

1 = 65°

So measure of ∠1 in the diagram is 65°.ANSWER

GUIDED PRACTICE for Examples 3 and 4

SOLUTION

A + B + C = 180°

x + 2x + 3x = 180°

6x = 180°

x = 30°

B = 2x = 2(30) = 60°

C = 3x = 3(30) = 90°

x

2x 3x

4. Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°.°

GUIDED PRACTICE for Examples 3 and 4

5. Find the measures of the acute angles of the right triangle in the diagram shown.

SOLUTION

Use the corollary to set up & solve an equation.

Corollary to the Triangle Sum Theorem(x – 6)° + 2x° = 90°

3x = 96

Solve for x.x = 32

Substitute 32 for x in equation x – 6 = 32 – 6 = 26°.

So, the measure of acute angle 2(32) = 64°ANSWER

GUIDED PRACTICE for Examples 3 and 4

6. In Example 4, what is the measure of the obtuse angle formed between the staircase and a segment extending from the horizontal leg?

A

B C Q

2x

xSOLUTION

First, sketch a diagram of the situation. Let the measure of the smaller acute angle be x° . Then the measure of the larger acute angle is 2x° . The Corollary to the Triangle Sum Theorem states that the acute angles of a right triangle are complementary.

GUIDED PRACTICE for Examples 3 and 4

Use the corollary to set up and solve an equation.

Corollary to the Triangle Sum Theoremx° + 2x = 90°

Solve for x.x = 30

So the measures of the acute angles are 30° and 2(30°) = 60°

ACD is linear pair to ACD.

So 30° + ACD = 180°.

Therefore = ACD = 150°.ANSWER