4.1

Warm UpWarm Up

Lesson QuizLesson Quiz

Lesson PresentationLesson Presentation

Apply Triangle Sum Properties

4.1 Warm-Up

1. 90º

ANSWER right

2. 72º

Classify each angle as acute, obtuse, or right.

ANSWER acute

3. 116º

ANSWER obtuse

4.1 Warm-Up

4. How do you know that 1 = 2?~

2

1

ANSWER Alt. Int. s Thm.

4.1 Example 1

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.

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

Support Beams

4.1 Example 2

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

4.1 Example 2

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

4.1 Guided Practice

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

obtuse isosceles triangle

B

A C

acute scalene triangleP

Q

R

SAMPLE ANSWER

4.1 Guided Practice

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.

isosceles; right triangle

ANSWER

4.1

Find mJKM.

Example 3

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 mJKM.

2x – 5 = 2 75 – 5 = 145

The measure of JKM is 145°.ANSWER

4.1

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.

Example 4

ARCHITECTURE

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.

4.1 Example 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°.

ANSWER

4.1 Guided Practice

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

The measure of 1 in the diagram is 65°.ANSWER

4.1 Guided Practice

4. Find the measure of each interior angle of ABC, where mA = x°, mB = 2x°, and mC = 3x°.

mA = 30°, mB = 60°, mC = 90°ANSWER

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

26° and 64°ANSWER

4.1 Guided Practice

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

150°ANSWER

4.1

1. Graph ABC with vertices A(0, 6), B(– 4, – 1) andC(4, – 1). Classify it by its sides. Then determine if it is a right triangle.

Lesson Quiz

isosceles; not a right angleANSWER

4.1 Lesson Quiz

2. Find x. Then classify the triangle by its angles.

22; acute ANSWER

3. Find the measure of the exterior angle shown.

104°ANSWER

4.1 Lesson Quiz

4. Find x and y.

82, 58 ANSWER