1. Angles of Triangles
Section 4.2

2. Objectives
Find angle measures in triangles.

3. Key Vocabulary
Corollary
Exterior angles
Interior angles

4. 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
Exterior Angle
Exterior Angle
Extend sides
Interior Angle

5. Triangle Interior and Exterior Angles
Smiley faces are interior angles and hearts represent the exterior angles

6. Triangle Interior and Exterior Angles))
))) ( A B C
Interior Angles
Exterior Angles (formed by extending the sides) ( )) (( D E F

7. Triangle Sum Theorem
The Triangle Angle-Sum Theorem gives the relationship among the interior angle measures of any triangle.

8. Theorem 4.1 – Triangle Sum Theorem
The sum of the measures of the angles of a triangle is 180°.
mX + mY + mZ = 180° X Y Z

9. Triangle Sum Theorem

10. Example 1 Given mA = 43° and mB = 85°, find mC. SOLUTION
mA + mB + mC = 180°
Triangle Sum Theorem
43° + 85° + mC = 180°
Substitute 43° for mA and 85° for mB.
128° + mC = 180°
Simplify.
128° + mC – 128° = 180° – 128°
Subtract 128° from each side.
mC = 52°
Simplify.
ANSWER
C has a measure of 52°.
CHECK
Check your solution by substituting 52° for mC. 43° + 85° + 52° = 180° 10

11. Example 2a A. Find p in the acute triangle. 73° + 44° + p° = 180°
Triangle Sum Theorem
73° + 44° + p° = 180°
117 + p = 180

12. Example 2b B. Find m in the obtuse triangle. 23° + 62° + m° = 180°
62
Triangle Sum Theorem
23° + 62° + m° = 180°
23 m

13. Your Turn: A. Find a in the acute triangle. Triangle Sum Theorem
38° a°
88°

14. Your Turn: B. Find c in the obtuse triangle. 24° + 38° + c° = 180°
Triangle Sum Theorem.
24° + 38° + c° = 180° c°
24°
38°

15. Now do these: a b c x y z r q p a = b = c = p = 180 – (90+57) = 33
Properties of Triangles
Now do these:
30o
80o a
41o
54o b
62o
34o c
141o x y z
58o
79o r q p
57o
a =
b =
c =
p = 180 – (90+57) = 33
x = 180 – 141 = 39
(vertically opposite
angles are equal)
y = 180 – (58+39)
= 83
q = 57
r = 180 – (79+57) = 44
z = 180 – 83 = 97

16. Example 3 Find the angle measures in the scalene triangle.
2x° + 3x° + 5x° = 180°
Triangle Sum Theorem
10x = 180
Simplify.
Divide both sides by 10.
x = 18
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°.

17. Your Turn: Find the angle measures in the scalene triangle.
3x° + 7x° + 10x° = 180°
Triangle Sum Theorem
The angle labeled 3x° measures _____ = __°,
the angle labeled 7x° measures ____= ___°, and the angle labeled 10x° measures _____ = ___°.
10x°
3x°
7x°

18. Example 4: 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.

19. Example 4: Angle Sum Theorem Simplify. Subtract 142 from each side.
Answer:

20. Your Turn:
Find the missing angle measures.
Answer:

21. Triangle Angle-Sum Corollaries
Corollary 4.1 – The acute s of a right ∆ are complementary. Example: m∠x + m∠y = 90˚ x° y°

22. Example 5 ∆ABC and ∆ABD are right triangles. Suppose mABD = 35°.
Find mDAB. a. b.
Find mBCD.
SOLUTION
Corollary to the
Triangle Sum Theorem
mDAB + mABD = 90° a.
mDAB + 35° = 90°
Substitute 35° for mABD.
mDAB + 35° – 35° = 90° – 35°
Subtract 35° from each side.
mDAB = 55°
Simplify.
Corollary to the
Triangle Sum Theorem
mDAB + mBCD = 90° b.
55° + mBCD = 90°
Substitute 55° for mDAB.
mBCD = 35°
Subtract 55° from each side. 22

23. Your Turn: 1. Find mA. 65° 2. Find mB. 75° Find mC. 50° ANSWER3.
ANSWER
50°

24. Example 6:
GARDENING The flower bed shown is in the shape of a right triangle. Find if is 20.
Corollary 4.1
Substitution
Subtract 20 from each side.
Answer:

25. Your Turn:
The piece of quilt fabric is in the shape of a right triangle. Find if is 62.
Answer:

26. 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.

27. Assignment
IXL W6
IXL W11
IXL W 12
IXL W 15