1. Angles of Triangles

2. Objectives
Find angle measures in triangles.

3. Key Vocabulary Supplementary Angles Complementary Angles
Exterior angles
Interior angles

4. Additional Vocabulary
Theorem - Mathematics. a theoretical proposition, statement, or formula embodying something to be proved from other propositions or formulas. 2. a rule or law, especially one expressed by an equation or formula. (Dictionary.com)

5. Theorems
4.1 Triangle Sum Theorem
4.2 Exterior Angle Theorem

6. 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

7. Triangle Interior and Exterior Angles
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.

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

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

10. 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.

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

12. 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

13. Triangle Sum Theorem

14. 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° 14

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

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

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

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

19. 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°.

20. Your Turn: Find the angle measures in the scalene triangle.
3x° + 7x° + 10x° = 180°
Triangle Sum Theorem
20x = 180
Simplify.
Divide both sides by 20.
x = 9
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°.
10x°
3x°
7x°

21. 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.

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

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

24. Example 5 ∆ABC and ∆ABD are right triangles. Suppose mABD = 35°.
Find mDAB. a. b.
Find mBCD.
SOLUTION
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.
mDAB + mBCD = 90° b.
55° + mBCD = 90°
Substitute 55° for mDAB.
mBCD = 35°
Subtract 55° from each side. 24

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

26. 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:

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

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

29. Exterior Angles and Triangles
An exterior angle is formed by one side of a triangle and the extension of another side (Example 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 3 4

30. 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 3 4

31. Example 7 Given mA = 58° and mC = 72°, find m1. SOLUTION
m1 = mA + mC
Exterior Angle Theorem
Substitute 58° for mA and 72° for mC.
= 58° + 72°
Simplify.
= 130°
ANSWER
1 has a measure of 130°. 31

32. Your Turn: 1. Find m2. 120° 2. Find m3. 155° 3. Find m4. 113°
ANSWER
120°
Find m3. 2.
ANSWER
155°
Find m4. 3.
ANSWER
113°

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

34. Example 8: Exterior Angle Theorem Substitution
Subtract 64 from each side.
If 2 s form a linear pair, they are supplementary.
Substitution
Simplify.
Subtract 78 from each side.

35. Example 8: Angle Sum Theorem Substitution Simplify.
Subtract 143 from each side.
Answer:

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

37. Success Starter
Solve for x:

38. Unit 6 Parallel Lines Learn about parallel line relationships
Prove lines parallel
Describe angle relationship in polygons

39. Parallel Lines Skew lines are non-coplanar, non-intersecting lines.
Coplanar lines that do not intersect. m
m || n n
Skew lines are non-coplanar, non-intersecting lines.
The only difference between skew and parallel is the coplanar part. What are the other ways to define parallel? p q

40. The Transversal Any line that intersects two or more coplanar lines. t

41. Lecture 2 Objectives Learn the special angle relationships …when lines
are parallel
                                    
Pictured: The Georges Pompidou Center.
Paris, France

42. When parallel lines are cut by a transversal…1 2 3 4 5 6 7 8
Corresponding ’s 
1   5
Alternate Interior ’s 
 4   5
Same Side Interior ’s Suppl.
 4 suppl.  6
This slide introduces the following three slides, which cover the postulates and theorems described here. Have them come up with other pairs of angles that match the conditions and are congruent.

43. r s t 1 2 3 4 5 6 7 8
If two parallel lines are cut by a transversal, then corresponding angles are congruent.
Postulate 10 cannot be proven, because it is the first angle relationship and so is assumed to be true. The other angle relationships can be proven, because of the assumption that corresponding angles are congruent. Either of the other two angle pairs could have been selected to serve as the assumption, and the others proven from that…the key here is that the choice was arbitrary.

44. r s t 1 2 3 4 5 6 7 8
If two parallel lines are cut by a transversal, then alternate interior angles are congruent.
Prove theorem 3-2 with them.

45. r s t 1 2 3 4 5 6 7 8
If two parallel lines are cut by a transversal, then same side interior angles are supplementary.
Ditto. This proof is a little harder, and is a good opportunity to see who is understanding proofs. How many ways are there to prove angles are supplementary? Investigate this first, as the answer leads the way.

46. A line perpendicular to one of two parallel lines is perpendicular to the other.
No need to prove this one. s

47. If two lines are cut by a transversal so that corresponding angles are congruent, then the lines are parallel.
If 1  2, then
m || n.
This postulate is the converse of post. 10 Since the direction is reversed, this postulate can be used to prove lines are parallel. All the statements today can be used to prove lines are parallel. 1 m 2 n

48. If two lines are cut by a transversal so that alternate interior angles are congruent, then the lines are parallel.
If 1  2, then
m || n.
Prove this with them. Note that none of the stuff learned yesterday can be used because they all start “in two parallel lines are cut by a transversal…” and in these proofs, the lines are not parallel, YET. Make sure to emphasize this! m 1 2 n

49. If two lines are cut by a transversal so that same side interior angles are supplementary, then the lines are parallel.
If 1 suppl  2, then
m || n.
Ditto. 1 m 2 n

50. In a plane, two lines perpendicular to the same line are parallel.
If t  m and t  n , then
m || n. t
No proof needed on this one. m n

51. Two lines parallel to the same line are parallel to each other
If p  m and m  n, then
p  n
This is the final way to prove lines are parallel. The next slide summarizes the ways. m n

52. Ways to Prove Lines are Parallel
Corresponding angles are congruent
Alternate interior angles are congruent
Same side interior angles are supplementary
In a plane, that two lines are perpendicular to the same line
Both lines are parallel to a third line