1. Relationships within Triangles
Chapter 5
Relationships within Triangles

2. 5-1 Objectives
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3. Midsegments
Definition of a midsegment: a segment connecting the __________________ of two sides of the triangle.
Triangle Midsegment Theorem: if a segment joins the midpoints of a triangle, then the segment is _____________ to the third side and is ___________ as long.

4. Identifying Parallel Segments
In ∆𝑋𝑌𝑍, A is the midpoint of XY, B is the midpoint of YZ, and C is the midpoint of ZX.
Draw a diagram to illustrate the problem.
What are the three pairs of parallel segments?

5. Using Properties of Midsegments
In the figure below, AD = 6 and DE = 7.5. What are the lengths of:
DC =
AC =
EF =
AB =

6. Using Coordinate Geometry - Midsegments
The coordinates of the vertices of a triangle are E(1, 2), F(5, 6), and G(3, -2).
Find the coordinates of H, the midpoint of EG, and J, the midpoint of FG.
Show that HJ  EF
Show that HJ = ½ EF.

7. Practice: pp. 289 – 290 #26, 38-42 Challenge: p. 291 #48

8. 5-2 Objectives
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9. Perpendicular Bisectors
Perpendicular Bisector Theorem: If a point is on the perpendicular bisector of a segment, then it is _____________________ from the ____________________ of the segment.
Converse of the Perpendicular Bisector Theorem: If a point is _________________ from the endpoints of a segment, then it is on the ________________________ of the segment.
If PM is the perpendicular bisector of AB, then _____ _____.

10. Using the Perpendicular Bisector Theorem
Find the length of QR.

11. Angle Bisectors
Angle Bisector Theorem: If a point is on the bisector of an angle, then the point is _________________ from the sides of the angle.
Converse of the Angle Bisector Theorem: If a point in the interior of an angle is equidistant from the sides of the angle, then the point is on the _________________.
If QS is the angle bisector of PQR, then ____  _____.

12. Using the Angle Bisector Theorem
What is the length of FB?

13. Practice: pp #9-11, 16, 17, 32, 33

14. 5-3 Objectives
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15. Definitions
Concurrent: Three or more lines that __________________________ ___________________________
Point of Concurrency: The point at which three lines ______________
In the diagram, the three lines are ____________________ and ________ is the point of concurrency. A

16. Concurrency of Perpendicular Bisectors Theorem
The perpendicular bisectors of the sides of a triangle are __________________ at a point equidistant from the _____________.
P is the _____________________ of the triangle. It is the center of the circle that is circumscribed about the triangle
In ∆ABC, _____  _____  _____

17. Location of the Circumcenter
Draw in the perpendicular bisectors of each triangle to find the location of the circumcenter for each type of triangle.
Acute Triangle
Right Triangle
Obtuse Triangle

18. Using Coordinate Geometry – Perpendicular Bisectors
What are the coordinates of the circumcenter of the triangle with vertices P(0, 6), O(0, 0), and S(4, 0)?

19. Concurrency of Angle Bisectors Theorem
The angle bisectors of a triangle are _________________ at a point equidistant from the ___________________________.
P is the _________________ of the triangle. It is the center of the circle that is inscribed in the triangle.
In ∆ABC, _____  _____  _____

20. Identifying and Using the Incenter
QN = 5x + 36 and QM = 2x What is QO?

21. Practice: pp # 11, 14, 16, 18

22. 5-4 Objectives
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23. Medians
A median of a triangle is a segment whose endpoints are a _______________ and the ___________________ of the opposite side.

24. Concurrency of Medians Theorem
The medians of a triangle are concurrent at a point that is ____________________ the distance from each _______________ to the _________________ of the opposite side.
C is the ______________ of the triangle. This is also called the center of gravity of a triangle because it is the point where a triangular shape will balance.
In ∆DEF, DJ, EG, and FH are medians.
DC = ____ EC = ____ FC = ____

25. Finding the Length of a Median
In ∆XYZ, ZA = 9.
What is the length of CA?
What is the length of ZC?

26. Altitudes
An altitude of a triangle is the __________________ segment from a _______________ of the triangle to the line containing the ______________________.

27. Concurrency of Altitudes Theorem
The lines that contain the altitudes of a triangle are concurrent.
H is the ________________ of the triangle.

28. Location of the orthocenter
Draw in the altitudes of each triangle to find the location of the orthocenter for each type of triangle.
Acute Triangle
Right Triangle
Obtuse Triangle

29. Using Coordinate Geometry - Altitudes
∆DEF has vertices D(1, 2), E(1, 6), and F(4, 2). What are the coordinates of the orthocenter of ∆DEF?

30. Identifying Medians and Altitudes
Determine whether each line is a median, altitude, or neither. EG AD CF

31. Naming Medians and Altitudes
A median in ∆ABC
An altitude for ∆ABC
A median in ∆AHC
An altitude for ∆AHB
An altitude for ∆AHG

32. Summary of Triangle Segments
Name the type of segments that are concurrent at the given point.

33. Practice: pp #8-10, 15, 17-20, 24-27

34. 5-5 Objectives
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35. Indirect Proof
A proof involving ______________ reasoning is an indirect proof (sometimes called proof by ________________).
Step 1: Temporarily assume the ______________ of what you want to prove (negation).
Step 2: Show that this temporary assumption leads to a _________________.
Step 3: Conclude that the temporary assumption must be ___________ and that what you want to prove must be true.

36. Writing an Indirect Proof
Given: ∆ABC is scalene (all sides have different lengths)
Prove: A, B, and C all have different measures
Step 1: Assume temporarily the opposite of what you want to prove.
Step 2: Show that this assumption leads to a contradiction
Step 3: Conclude that the temporary assumption must be false and that what you want o prove must be true.

37. Writing an Indirect Proof
Given: 7 𝑥+𝑦 =70 𝑎𝑛𝑑 𝑥≠4
Prove: 𝑦≠6
Step 1: Assume temporarily the opposite of what you want to prove.
Step 2: Show that this assumption leads to a contradiction
Step 3: Conclude that the temporary assumption must be false and that what you want o prove must be true.

38. Writing an Indirect Proof
Given: ∆LMN
Prove: ∆LMN has at most one right angle

39. Practice: pp #17, 23, 24, 29

40. 5-6 Objectives
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41. Theorems
Theorem 5-10: If two sides of a triangle are not congruent, then the _____________ angle lies opposite the ____________ side.
Theorem 5-11: If two angles of a triangle are not
congruent, then the ____________ side lies
opposite the ____________ angle.
Triangle Inequality Theorem: The sum of the lengths of any two sides of a triangle is _____________________ the length of the third side.

42. Using Theorem 5-10 and 5-11
In ∆SOX, OX = 4, OS = 6, and XS = 9. Order the angles from least to greatest.
In ∆SOX, mS = 24 and mO = 130. Which side of ∆SOX is the shortest side?

43. Using the Triangle Inequality Theorem
Can a triangle have sides with lengths 2 m, 6m, and 9m? Explain.
A triangle has side lengths of 4 in. and 7 in. What is the range of possible lengths for the third side?

44. Practice: pp #12, 19, 25, 26, 30, 31, 37-39

45. 5-7 Objectives
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46. The Hinge Theorem (SAS Inequality Theorem)
If two sides of one triangle are congruent to two sides of another triangle, and the included angles are not congruent, then the longer third side is opposite the larger included angle.
BC ______ YZ
Converse of the Hinge Theorem (SSS Inequality): If two sides of one triangle are congruent to two sides of another triangle, and the third sides are not congruent, then the larger included angle is opposite the longer third side.
A ______ X

47. Using the Hinge Theorem
What inequality relates LN and OQ in the figure below.

48. Using the Converse of the Hinge Theorem
What is the range of possible values for x in the figure below?

49. Practice: pp #6-9, 11-14, 16-18