1. Angles of Triangles
3-4

2. EXAMPLE 1
Classify triangles by sides and by angles
Support Beams
Classify the triangular shape of the support beams in the diagram by its sides and by measuring its 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.

3. Classify a triangle in a coordinate plane
EXAMPLE 2
Classify a triangle in a coordinate plane
Classify PQO by its sides. Then determine if the triangle is a right triangle.
SOLUTION
STEP 1
Use the distance formula to find the side lengths. OP = y 2 – 1 ( ) x + = 2 – ( )
(– 1 ) + 5
2.2 OQ = y 2 – 1 ( ) x + 2 = – ( ) 6 + 3 45
6.7

4. Classify a triangle in a coordinate plane
EXAMPLE 2
Classify a triangle in a coordinate plane PQ = y 2 – 1 ( ) x + 3 – 2 ( ) 6 + =
(– 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 = 2 1 . 1
The 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

5. GUIDED PRACTICE
for Examples 1 and 2
Draw an obtuse isosceles triangle and an acute scalene triangle.
obtuse isosceles triangle B A C
acute scalene triangle P Q R

6. Use the distance formula to find the side lengths.
GUIDED PRACTICE
for Examples 1 and 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 = y 2 – 1 ( ) x + = 3 – ( ) 2
( 3 ) + = 18
4.2 BC = y 2 – 1 ( ) x + 2 =
– 3 ( ) –3 3 + – =
400 20

7. The product of the slopes is 1(– 1) = – 1 ,
GUIDED PRACTICE
for Examples 1 and 2 AC = y 2 – 1 ( ) x + = 3 – ( ) 2
0 )
(–3 + 18
4.2
STEP 2
Check for right angles.
The slope of AB is
3 – 0 = 1.
The slope of AC is
3 – 0
– 3 – 0 = .
– 1
The product of the slopes is
1(– 1) =
– 1 ,
so AB AC and BAC is a right angle.
Therefore, ABC is a right Isosceles triangle.
ANSWER

8. Write and solve an equation to find the value of x.
EXAMPLE 3
Find an angle measure
ALGEBRA
Find m∠ JKM.
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
(2x – 5)° =
70° + x°
Apply the Exterior Angle Theorem.
x = 75
Solve for x.
STEP 2
Substitute 75 for x in 2x – 5 to find m∠ JKM.
2x – 5 = 2 75
– 5 =
145
The measure of ∠ JKM is 145°.
ANSWER

9. 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.

10. Find angle measures from a verbal description
EXAMPLE 4
Find angle measures from a verbal description
Use the corollary to set up and solve an equation.
x° + 2x° =
90°
Corollary to the Triangle Sum Theorem
x = 30
Solve for x.
So, the measures of the acute angles are 30° and 2(30°)
= 60° .
ANSWER

11. Find the measure of 1 in the diagram shown.
GUIDED PRACTICE
for Examples 3 and 4
Find the measure of 1 in the diagram shown.
SOLUTION
STEP 1
Write and solve an equation to find the value of x.
(5x – 10)° =
40° + 3x°
Apply the Exterior Angle Theorem.
2x = 50
Solve for x. x= 25

12. GUIDED PRACTICE
for Examples 3 and 4
STEP 2
Substitute 25 for x in 5x – 10 to find
5x – 10 = 5 25
– 10 =
115
1 + (5x – 10)° =
180
° =
180°
1 =
65°
So measure of ∠ 1 in the diagram is 65°.
ANSWER

13. GUIDED PRACTICE
for Examples 3 and 4 x 2x 3x
Find the measure of each interior angle of ABC, where m A = x , m B = 2x° , and m C = 3x°.
SOLUTION
A + B C =
180°
x + 2x + 3x =
180°
6x =
180°
x =
30°
B =
2x =
2(30) =
60°
C =
3x =
3(30) =
90°

14. Use the corollary to set up & solve an equation.
GUIDED PRACTICE
for Examples 3 and 4
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.
(x – 6)° + 2x° =
90°
Corollary to the Triangle Sum Theorem
3x = 96
x = 32
Solve for x.
Substitute 32 for x in equation x – 6 =
32 – 6 =
26°.
So, the measure of acute angle 2(32) =
64°
ANSWER

15. GUIDED PRACTICE for Examples 3 and 4
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 x
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.

16. Use the corollary to set up and solve an equation.
GUIDED PRACTICE
for Examples 3 and 4
Use the corollary to set up and solve an equation.
x° + 2x =
90°
Corollary to the Triangle Sum Theorem
x = 30
Solve for x.
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