1. Classify each angle as acute, obtuse, or right.
1. 90º
ANSWER
right
2. 72º
ANSWER
acute
3. 116º
ANSWER
obtuse
4. How do you know that 1 = 2? ~ 2 1
ANSWER
Alt. Int. s Thm.

2. Classify Triangles by Sides and Angles
Target
Classify Triangles by Sides and Angles
You will…
Classify triangles and find measures of their angles.

3. Vocabulary
triangle – a polygon with 3 sides
To classify by sides:
scalene – no congruent sides
isosceles – at least 2 congruent sides
equilateral – exactly 3 congruent sides
To classify by angles:
acute – exactly 3 acute angles
right – exactly 1 right angle
obtuse – exactly 1 obtuse angle
equiangular – exactly 3 congruent angles

4. Vocabulary
interior angles – original angles of the triangle – “inside”
exterior angles – the angles that form linear pairs with the interior angles of a triangle

5. Vocabulary
Triangle Sum Theorem 4.1 –
the sum of the measures of the interior angles of any triangle is 180°
Corollary – the acute angles of a right triangle are complementary

6. Vocabulary
Exterior Angle Theorem 4.2 –
the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles
m A + m B = m 1

7. 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.

8. 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 PQ = y 2 – 1 ( ) x + 3 – 2 ( ) 6 + =
(– 1 ) 50
7.1

9. EXAMPLE 2
Classify a triangle in a coordinate plane
STEP 2
Check for right angles.
The slope of OP is
2 – 0
– 1– 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

10. 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

11. ABC is a right Isosceles triangle.
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 right. = 3 – ( ) 2
( 3 ) + 18 AB
ABC
is isosceles = −3 – 3 ( ) 2
( 3 ) + 36 BC = – 3 ( ) 2
( 0 ) + 18 AC
The slope of AB is
3 – 0 = 1.
BAC
is a right angle
The slope of BC is
3 – 3
−3 – 3 = 0.
The slope of AC is
3 – 0
−3 – 0 =
−1.
ANSWER
ABC is a right Isosceles triangle.

12. Write and solve an equation to find the value of x.
EXAMPLE 3
Find an angle measure
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

13. 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.

14. 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

15. GUIDED PRACTICE
for Examples 3 and 4
Find the measure of 1 in the diagram shown.
The measure of ∠ 1 in the diagram is 65°.
ANSWER

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

17. GUIDED PRACTICE for Examples 3 and 4
Find the measures of the acute angles of the right triangle in the diagram shown.
26° and 64°
ANSWER
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
ACD =150°.
ANSWER