1. November 7, 2018
5.1 Angles of Triangles

2. Geometry
5.1 Angles of Triangles

3. Essential Question How are the angle measures of a triangle related?
November 7, 2018
5.1 Angles of Triangles

4. Goals – Day 1 Classify triangles by their sides
Classify triangles by their angles
Identify parts of triangles.
Find angle measures in triangles.
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5.1 Angles of Triangles

5. Triangle Symbol Use the picture  for triangle. November 7, 2018
5.1 Angles of Triangles

6. This is ABC, which can also be named BCA, CAB, BAC, CBA, or ACB.
Triangle
A triangle is a figure formed by three segments joining three noncollinear points. B A C
This is ABC, which can also be named BCA, CAB, BAC, CBA, or ACB.
November 7, 2018
5.1 Angles of Triangles

7. Classifying Triangles by Sides
Equilateral 
Isosceles 
Scalene 
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5.1 Angles of Triangles

8. Equilateral Triangle Three congruent sides. November 7, 2018
5.1 Angles of Triangles

9. Isosceles Triangle At least two congruent sides. November 7, 2018
5.1 Angles of Triangles

10. Scalene Triangle No congruent sides. November 7, 2018
5.1 Angles of Triangles

11. Classifying Triangles by Angles
Right 
Equiangular 
Acute 
Obtuse 
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5.1 Angles of Triangles

12. Right Triangle One Right Angle November 7, 2018
5.1 Angles of Triangles

13. Equiangular Triangle Three Congruent Angles November 7, 2018
5.1 Angles of Triangles

14. Acute Triangle Three acute angles November 7, 2018
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15. Obtuse Triangle One Obtuse Angle November 7, 2018
5.1 Angles of Triangles

16. And to add to the confusion…
An equilateral triangle is also equiangular.
An equiangular triangle is also acute.
An equilateral can be considered an isosceles triangle.
An equilateral triangle is also acute.
November 7, 2018
5.1 Angles of Triangles

17. Adjacent and Opposite Sides of a Triangle
Two sides that share a common vertex are adjacent sides.
The third side is the opposite side from that vertex. A
In RAT, RA and RT are adjacent sides.
AT is the opposite side from ∠𝑅. R T
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5.1 Angles of Triangles

18. Isosceles Triangles
(In this case, we consider an isosceles triangle with only two congruent sides.)
The congruent sides are the LEGS.
The third side is the BASE.
Leg
Leg
Base
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5.1 Angles of Triangles

19. Right Triangle The LEGS form the right angle.
The third side (opposite the right angle) is the Hypotenuse.
Hypotenuse
Leg
Leg
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5.1 Angles of Triangles

20. Hypotenuse From the Greek “stretched against”.
Always longer than either leg.
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5.1 Angles of Triangles

21. What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M
1. Name the legs of the isosceles triangle PMQ.
Segments PM and QM.
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5.1 Angles of Triangles

22. What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M
2. Name the base of isosceles triangle  PMQ.
Segment PQ.
November 7, 2018
5.1 Angles of Triangles

23. What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M
3. Name the hypotenuse of right triangle PNM.
Segment PM.
November 7, 2018
5.1 Angles of Triangles

24. What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M
4. Name the legs of right triangle  PNM.
Segments NP and NM.
November 7, 2018
5.1 Angles of Triangles

25. What have you learned so far?
In the figure, 𝑀𝑁 ⊥ 𝑄𝑃 and 𝑀𝑃 ≅ 𝑀𝑄 . Complete the following sentence. P Q N M
5. Name the acute angles of right triangle  QNM.
Q and NMQ
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5.1 Angles of Triangles

26. Example 1 Classify these triangles by its angles and by its sides. a.
125°
Obtuse ,
Isosceles 
Equiangular, Equilateral 
Isosceles , Acute 
Right , Scalene 
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27. 5.1 Triangle Sum Theorem
The sum of the measures of the interior angles of a triangle is 180°. A B C
mA + mB + mC = 180°
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5.1 Angles of Triangles

28. Example 2 Find the measure of 1. Solution: m1 + 70 + 32 = 180
70°
32° 1
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5.1 Angles of Triangles

29. Example 3 In MAD: mM = (2x)° mA = (3x)° mD = (4x)
Find the measure of each angle, and classify.
Solution:
2x + 3x + 4x = 180
9x = 180
x = 20
= 2(20) = 40°
= 3(20) = 60°
= 4(20) = 80°
This triangle is acute.
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5.1 Angles of Triangles

30. Example 4 In RST: mR=(5x + 10) mS=(2x + 15) mT=(3x + 35)
Find the measure of the three angles and then classify the triangle by angles.
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5.1 Angles of Triangles

31. ACUTE Example 4 Solution (5x + 10) + (2x + 15) + (3x + 35) = 180
mR=(5x + 10) = 5(12) + 10 = 70
mS=(2x + 15) = 2(12) + 15 = 39
mT=(3x + 35) = 3(12) + 35 = 71
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5.1 Angles of Triangles

32. Corollary to Theorem 5.1
The acute angles of a right triangle are complementary.
m1 + m = 180
m1 + m2 = 90
QED 1 2
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5.1 Angles of Triangles

33. Example 5 Find X
x = 70°
Since this is a right triangle, the acute angles are complementary, and 90 – 20 = 70.
20° x°
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34. Interior and Exterior Angles
Start with a triangle…
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35. 1, 2, 3 are INTERIOR ANGLES. They are INSIDE the triangle.
Extend the sides…. 2 1 3
1, 2, 3 are INTERIOR ANGLES.
They are INSIDE the triangle.
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5.1 Angles of Triangles

36. 4, 6, 8, 9, 10, and 12 are EXTERIOR ANGLES. 3 10 6 12
4, 6, 8, 9, 10, and 12 are EXTERIOR ANGLES.
They are OUTSIDE the triangle.
They are ADJACENT to the interior angles.
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37. 5, 7, and 11 are NOT EXTERIOR ANGLES. 2 1 3 5 11
5, 7, and 11 are NOT EXTERIOR ANGLES.
They are simply vertical angles to the interior angles.
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5.1 Angles of Triangles

38. Exterior angles are always supplementary to the interior angles.
It is common (and less confusing) to draw only one exterior angle at a vertex.
Exterior angles are always supplementary to the interior angles. 6 3 1 2 5 4
Interior Angles: 1, 2, 3
Exterior Angles: 4, 5, 6
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5.1 Angles of Triangles

39. 5.2 Exterior Angle Theorem
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles. 1 2 3
m1 = m2 + m3
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5.1 Angles of Triangles

40. Note:
Sometimes (usually) the two nonadjacent interior angles are referred to as REMOTE INTERIOR ANGLES. The theorem then reads:
An exterior angle of a triangle is equal to the sum of the two remote interior angles.
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5.1 Angles of Triangles

41. 5.2 Exterior Angle Thm Proof (Informal)
m2 + m3 + m4 = 180 ( angle sum)
m4 + m1 = 180 (linear pair postulate)
m2 + m3 + m4 = m4 + m1 (substitution)
m2 + m3 = m1 (subtraction) 1 2 3 4
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5.1 Angles of Triangles

42. Naming Remote Interior Angles1 2 3 4 6 5 8 7 9
For exterior 1, the remote interior angles are_____________.
𝒎𝟔 + 𝒎𝟖=𝒎1
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5.1 Angles of Triangles

43. Naming Remote Interior Angles1 2 3 4 6 5 8 7 9
For exterior 4, the remote interior angles are_____________.
𝒎𝟐 + 𝒎𝟖=𝒎𝟒
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5.1 Angles of Triangles

44. Naming Remote Interior Angles1 2 3 4 6 5 8 7 9
For exterior 5, the remote interior angles are_____________.
𝒎𝟐 + 𝒎𝟖=𝒎𝟓
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5.1 Angles of Triangles

45. Naming Remote Interior Angles1 2 3 4 6 5 8 7 9
For exterior 9, the remote interior angles are_____________.
𝒎𝟔 + 𝒎𝟐=𝒎𝟗
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46. Naming Remote Interior Angles1 2 3 4 6 5 8 7 9
For remote interior angles 6 & 8, the exterior angle is _____________.
𝒎𝟔 + 𝒎𝟖=𝒎∠𝟏=𝒎∠𝟑
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5.1 Angles of Triangles

47. Naming Remote Interior Angles1 2 3 4 6 5 8 7 9
For remote interior angles 2 & 6, the exterior angle is _____________.
𝒎𝟔 + 𝒎𝟐=𝒎∠𝟕=𝒎∠𝟗
November 7, 2018
5.1 Angles of Triangles

48. Naming Remote Interior Angles1 2 3 4 6 5 8 7 9
For remote interior angles 2 & 8, the exterior angle is _____________.
𝒎𝟐 + 𝒎𝟖=𝒎∠𝟒=𝒎∠𝟓
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5.1 Angles of Triangles

49. Example 6 Find m1. By Theorem 5.2: m1 + 45 = 110
45°
By Theorem 5.2:
m = 110
m1 = 110 – 45 = 65° 1
110°
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5.1 Angles of Triangles

50. Example 7 (x + 15) + 45 = 3x – 10 x + 60 = 3x – 10 70 = 2x x = 35
45°
(x + 15)°
(3x – 10)°
Solve for x.
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5.1 Angles of Triangles

51. A Final Challenge Problem…
Find the measure of each numbered angle.
40°
30°
60°
20° 1 2 3 4 5 6 7
50°
60°
60°
90°
60°
100°
60°
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5.1 Angles of Triangles

52. Problems for You Use the exterior angle theorem!
Write down the equation for each problem and solve.
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53. Your Turn. 1. Find m1 Solution: m1 = 32 + 125 m1 = 157 32 1 125
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54. 2. Find m2 Solution: m2 + 45 = 165 m2 = 120 45 2 165
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55. 3. Solve for x. Solution: 2x + 30 + 60 = 110 2x + 90 = 110 2x = 20
110°
(2x + 30)°
60
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56. 4. Solve for x. Solution: 12x – 4 = (6x + 8) + 5x 12x – 4 = 11x + 8
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57. 5. Solve for x. Solution: (3x + 2) + (5x – 10) = 7x + 3
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58. Summary The sum of the interior angles of a triangle is 180 degrees.
The acute angles of a right triangle are complementary.
An exterior angle is equal to the sum of the two remote interior angles.
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59. Assignment
November 7, 2018
5.1 Angles of Triangles