1. Relationships within Triangles
Chapter 6

2. Perpendicular and Angle Bisectors
I can use perpendicular and angle bisectors to find measures and distance relationships.

3. Perpendicular and Angle Bisectors
Vocabulary (page 166 in Student Journal)
equidistant: a point that is the same distance away from 2 objects

4. Perpendicular and Angle Bisectors
Core Concepts (pages 166 and 167 in Student Journal)
Perpendicular Bisector Theorem
In a plane, if a point lies on the perpendicular bisector of a segment, then it is equidistant from the endpoints of the segment.

5. Perpendicular and Angle Bisectors
Converse of the Perpendicular Bisector Theorem
In a plane, if a point is equidistant from the endpoints of a segment, then it lies on the perpendicular bisector of the segment.

6. Perpendicular and Angle Bisectors
Angle Bisector Theorem
If a point lies on the bisector of an angle, then it is equidistant from the 2 sides of the angle.
Converse of the Angle Bisector Theorem
If a point is in the interior of an angle and is equidistant from 2 sides of the angle, then it lies on the bisector of the angle.

7. Perpendicular and Angle Bisectors
Examples (space on pages 166 and 167 in Student Journal)
Find each measure.
a) CD b) PR c) GH

8. Perpendicular and Angle Bisectors
Solutions 27 10 16

9. Perpendicular and Angle Bisectors
Find each measure.
d) angle ABC e) JM

10. Perpendicular and Angle Bisectors
Solutions
d) 112 degrees e) 24

11. Perpendicular and Angle Bisectors
f) Write an equation of the perpendicular bisector of the segment with endpoints D(5, -1) and E(-11, 3).

12. Perpendicular and Angle Bisectors
Solution
f) y = 4x + 13

13. Bisectors in Triangles
I can use and find the circumcenter and incenter of a triangle.

14. Bisectors in Triangles
Vocabulary (page 171 in Student Journal)
concurrent: 3 or more lines intersect at the same point
point of concurrency: the point where 3 or more lines intersect

15. Bisectors in Triangles
circumcenter: the point of concurrency of the perpendicular bisectors of a triangle
incenter: the point of concurrency of the angle bisectors of a triangle

16. Bisectors of Triangles
Core Concepts (pages 171 and 172 in Student Journal)
Circumcenter Theorem
The perpendicular bisectors of the sides of a triangle are concurrent at a point (circumcenter) equidistant from the vertices.

17. Bisectors of Triangles
Incenter Theorem
The bisectors of the angles of a triangle are concurrent at a point (incenter) equidistant from the sides of the triangles.

18. Bisectors of Triangles
Examples (space on pages 171 and 172 in Student Journal)
a) Find the location of the food stand so that it is equidistant from all 3 rides.

19. Bisectors of Triangles
Solution
a) the circumcenter

20. Bisectors of Triangles
b) Find the coordinates of the circumcenter of triangle DEF with vertices D(6, 4), E(-2, 4) and F(-2, -2).

21. Bisectors of Triangles
Solutions
b) (2, 1)

22. Bisectors of Triangles
c) In the figure, NE = 6x + 1 and NF = 4x Find ND.

23. Bisectors of Triangles
Solution
c) 43

24. Bisectors of Triangles
d) A school has a fenced in playground the shape of a scalene triangle. The principal would like to play a swing center in the playground so that it is equidistant from each of the 3 fences. Where should the swing set be placed?

25. Bisectors of Triangles
d) the incenter

26. Medians and Altitudes of Triangles
I can use medians and find centroids of triangles and use altitudes and find the orthocenters of triangles.

27. Medians and Altitudes of Triangles
Vocabulary (page 176 in Student Journal)
median of a triangle: a segment connecting a vertex and the midpoint of the opposite side
centroid: the point of concurrency for the medians, which is also the point where the triangular shape will balance

28. Medians and Altitudes of Triangles
altitude of a triangle: the perpendicular segment connecting a vertex to the opposite side
orthocenter: the point of concurrency for the lines that contain the altitudes of a triangle

29. Medians and Altitudes of Triangles
Core Concepts (page 176 and 177 in Student Journal)
Centroid Theorem
The medians of a triangle are concurrent at a point (centroid) that is ⅔ the distance from each vertex to the midpoint of the opposite side

30. Medians and Altitudes of Triangles
Orthocenter Theorem
The lines that contain the altitudes of a triangle are concurrent at a point (orthocenter).

31. Medians and Altitudes of Triangles
Examples (space on pages 176 and 177 in Student Journal)
a) Point Q is the centroid of triangle RST. If VQ = 5, find RQ and RV.

32. Medians and Altitudes of Triangles
Solution
a) RQ = 10 and RV = 15

33. Medians and Altitudes of Triangles
b) Find the coordinates of the centroid of triangle ABC with vertices A(0, 4), B(-4, -2) and C(7,1).

34. Medians and Altitudes of Triangles
Solution
b) (1, 1)

35. Medians and Altitudes of Triangles
c) Find the coordinates of the orthocenter of triangle DEF with vertices D(0, 6), E(-4, -2) and F(4, 6).

36. Medians and Altitudes of Triangles
Solution
c) (-4, 10)

37. Medians and Altitudes of Triangles
d) Prove that the bisector of the vertex angle of an isosceles triangle is an altitude.

38. Medians and Altitudes of Triangles
Solution d)

39. The Triangle Midsegment Theorem
I can use the Triangle Midsegment Theorem to find distances.

40. The Triangle Midsegment Theorem
Vocabulary (page 181 in Student Journal)
midsegment of a triangle: a segment connecting 2 sides of a triangle at their midpoints

41. The Triangle Midsegment Theorem
Core Concepts (page 181 in Student Journal)
Triangle Midsegment Theorem
If a segment joins the midpoint of 2 sides of a triangle, then the segment is parallel to the third side and half as long.

42. The Triangle Midsegment Theorem
Examples (space on page 181 in Student Journal)
In triangle RST, show that midsegment MN is parallel to segment RS and MN = 1/2 RS.

43. The Triangle Midsegment Theorem
Solution a)

44. The Triangle Midsegment Theorem
b) Segment DE is the midsegment of triangle ABC. Find AC.

45. The Triangle Midsegment Theorem
Solution
b) AC = 9.6

46. Indirect Proof and Inequalities in One Triangle
I can use the Triangle Inequality Theorem to find possible side lengths of triangles.

47. Indirect Proof and Inequalities in One Triangle
Vocabulary (page 186 in Student Journal)
indirect proof: make the assumption that the desired conclusion is false and then show this assumption is logically impossible to prove the original statement true by contradiction

48. Indirect Proof and Inequalities in One Triangle
Core Concepts (pages 186 and 187 in Student Journal)
Triangle Longer Side Theorem
If 1 side of a triangle is longer than another side, the angle opposite the longer side is larger than the angle opposite the shorter side.

49. Indirect Proof and Inequalities in One Triangle
Triangle Larger Angle Theorem
If 1 angle of a triangle is larger than another angle, the side opposite the larger angle is longer than the side opposite the smaller angle.

50. Indirect Proof and Inequalities in One Triangle
Triangle Inequality Theorem
The sum of the lengths of any 2 sides of a triangle is greater than the length of the third side.

51. Indirect Proof and Inequalities in One Triangle
Examples (space on pages 186 and 187 in Student Journal)
a) Write an indirect proof. Given that line l is not parallel to line k, prove angle 3 and angle 5 are not supplementary.

52. Indirect Proof and Inequalities in One Triangle
Solution a)

53. Indirect Proof and Inequalities in One Triangle
b) List the angles of triangle ABC in order from least to greatest.

54. Indirect Proof and Inequalities in One Triangle
Solution
b) angle A, angle B, angle C

55. Indirect Proof and Inequalities in One Triangle
c) List the sides of triangle ABC in order from greatest to least.

56. Indirect Proof and Inequalities in One Triangle
Solution
c) segment AC, segment AB, segment BC

57. Indirect Proof and Inequalities in One Triangle
d) A triangle has 1 side length of 6 units and another side length of 15 units. Describe the possible lengths of the 3rd side.

58. Indirect Proof and Inequalities in One Triangle
Solution
d) 9 < s < 21

59. Inequalities in Two Triangles
I can solve problems using the Hinge Theorem.

60. Inequalities in Two Triangles
Core Concepts (page 191 in Student Journal)
The Hinge Theorem
If the 2 sides of one triangle are congruent to the 2 sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger of the included angles.

61. Inequalities in Two Triangles
Converse of the Hinge Theorem
If 2 sides of 1 triangle are congruent to 2 sides of another triangle, and the 3rd side of the 1st is longer than the 3rd side of the 2nd, then the included angle of the 1st is larger than the included angle of the 2nd.

62. Inequalities in Two Triangles
Examples (space on page 191 in Student Journal)
a) Based on the diagram, how does the measure of angle B compare to the measure of angle E?

63. Inequalities in Two Triangles
Solution
a) measure of angle B > measure of angle E

64. Inequalities in Two Triangles
b) Based on the diagram, how does AC compare to DC?

65. Inequalities in Two Triangles
Solution
b) AC > DC

66. Inequalities in Two Triangles
c) Given AB = BC and AD > CD, prove the measure of angle ABD > measure of angle CBD.

67. Inequalities in Two Triangles
Solution c)

68. Inequalities in Two Triangles
d) Two groups of bikers leave the same camp heading in opposite directions. Each group travels 2 miles and then changes directions and travels another 1.2 miles. Group A starts east and turns 45 degrees towards the north. Group B starts west and travels 30 degrees toward the south. Which group is farther from camp?

69. Inequalities in Two Triangles
Solution
d) Group B