triangle sum theorem

1. Angles of Triangles
Section 4.2

2. Objectives
Find angle measures in triangles.

3. Key Vocabulary
Corollary
Exterior angles
Interior angles

4. Theorems
4.1 Triangle Sum Theorem
 Corollary to the Triangle Sum Theorem
4.2 Exterior Angle Theorem

5. 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
Extend sides
Interior
Angle
Exterior
Angle
Exterior
Angle

6. Triangle Interior and Exterior Angles
A
B
C
Smiley faces are interior
angles and hearts
represent the exterior
angles
Each vertex has a pair
of congruent exterior
angles; however it is
common to show only
one exterior angle at
each vertex.

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

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

9. Triangle Sum Theorem
If you tear off two corners of a triangle and
place them next to the third corner, the
three angles seem to form a straight line.
You can also show this in a drawing.

10. Draw a triangle and extend one side. Then
draw a line parallel to the extended side, as
shown.
The three angles in the triangle can be
arranged to form a straight line or 180°.
Two sides of the
triangle are
transversals to the
parallel lines.
Triangle Sum Theorem

11. 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

12. Triangle Sum Theorem

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

14. A. Find p in the acute triangle.
73° + 44° + p° = 180°
117 + p = 180
p = 63
–117 –117
Triangle Sum
Theorem
Subtract 117 from
both sides.
Example 2a

15. B. Find m in the obtuse triangle.
23° + 62° + m° = 180°
85 + m = 180
m = 95
–85 –85
Triangle Sum
Theorem
Subtract 85 from
both sides.
23
62
m
Example 2b

16. A. Find a in the acute triangle.
88° + 38° + a° = 180°
126 + a = 180
a = 54
–126 –126
88°
38°
a°
Triangle Sum
Theorem
Subtract 126
from both sides.
Your Turn:

17. B. Find c in the obtuse triangle.
24° + 38° + c° = 180°
62 + c = 180
c = 118
–62 –62
c°
24°
38°
Triangle Sum
Theorem.
Subtract 62 from
both sides.
Your Turn:

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

19. 3x° + 7x° + 10x° = 180°
20x = 180
x = 9
20 20
Find the angle measures in the scalene triangle.
Triangle Sum Theorem
Simplify.
Divide both sides by 20.
3x° 7x°
10x°The angle labeled 3x°
measures 3(9°) = 27°, the
angle labeled 7x°
measures 7(9°) = 63°, and
the angle labeled 10x°
measures 10(9°) = 90°.
Your Turn:

20. 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.
Example 4:

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

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

23. Corollaries
Definition: A corollary is a theorem with a
proof that follows as a direct result of
another theorem.
As a theorem, a corollary can be used as
a reason in a proof.

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

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

26. ANSWER 65°
ANSWER 75°
ANSWER 50°
Find mA.1.
Find mB.2.
Find mC.3.
Your Turn:

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

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

29. Exterior Angles and Triangles
An exterior angle is formed by one side of a
triangle and the extension of another side
(i.e. 1 ).
The interior angles of the triangle not adjacent to
a given exterior angle are called the remote
interior angles (i.e. 2 and 3).
1
2
34

30. Investigating Exterior Angles of a
Triangles
B
A
A
B
C
You can put the two torn angles
together to exactly cover one of the
exterior angles

31. Theorem 4.2 – Exterior Angle Theorem
The measure of an exterior angle of a
triangle is equal to the sum of the
measures of the two remote interior
angles.
m 1 = m 2 + m 3
1
2
34

32. ANSWER 1 has a measure of 130°.
SOLUTION
m1 = mA + mC Exterior Angle Theorem
Given mA = 58° and mC = 72°, find m1.
Substitute 58° for mA and
72° for mC.
= 58° + 72°
Simplify.= 130°
Example 7

33. ANSWER 120°
ANSWER 155°
ANSWER 113°
Find m2.1.
Find m3.2.
Find m4.3.
Your Turn:

34. Find the measure of each numbered angle in the figure.
Exterior Angle Theorem
Simplify.
Substitution
Subtract 70 from each side.
If 2 s form a linear pair, they
are supplementary.
Example 8:

35. Exterior Angle Theorem
Subtract 64 from each side.
Substitution
Subtract 78 from each side.
If 2 s form a linear pair,
they are supplementary.
Substitution
Simplify.
Example 8: m∠1=70
m∠2=110

36. Subtract 143 from each side.
Angle Sum Theorem
Substitution
Simplify.
Answer:
Example 8:
m∠1=70
m∠2=110
m∠3=46
m∠4=102

37. Find the measure of each numbered angle in the figure.
Answer:
Your Turn:

38. 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.

39. Assignment
Pg. 182-184: #1 – 13 odd, 19 – 29 odd