Chapter 6 Relationships Triangles
Name: _______________
A triangular roof truss is to be created according to
the diagram. The king post is constructed in the
center of the bottom chord. What conclusions
can you make about the roof lines as the king
post gets longer? What conclusions can you
make about the two top chords and the angles they form?
6.1
Warm Up
The diagram includes a pair of congruent triangles.
Use the congruent triangles to find the value of x in the diagram.
1.
3.
2.
4.
105
Section 6.1-Perpendicular and Angle Bisectors
106
6.1
Practice A
Tell whether the information in the diagram allows you to conclude that point P lies on the perpendicular
bisector of RS, or on the angle bisector of DEF. Explain your reasoning.
1.
2.
3.
Find the indicated measure. Explain your reasoning.
4. AD
5. GJ
6. PQ
7. mDGF
8.AC
9. mLNM
10.
mUTW
107
Cumulative Review Warm-Up
Graph the polygon and its image after a reflection in the given line.
1.
y  x
2.
y  2
Find the measure of the unknown angle(s) in the triangle.
1.
2.
Find the value of x that makes 𝒎 ∥ 𝒏
1.
2.
3.
108
Lines of Concurrency
Define:
1) Concurrent lines2) Point of concurrency-
Definition
Angle Bisector
A bisector that divides
an angle of a triangle
into two congruent
adjacent angles
Perpendicular
A segment that is
perpendicular to a side
of a triangle at the
midpoint of the side
Bisector
Median
Altitude
Point of
concurrency
Drawing
A segment whose
endpoints are a vertex
of a triangle and the
midpoint of the
opposite side
The perpendicular
segment from a vertex
of a triangle to the
opposite side or to the
line that contains the
opposite side
Identify the following:
1)
______________
2)
______________
3)
_____________
4)
____________
109
Points of Concurrency
Name__________________________
Given the following pictures and markings, identify each of the following as (a) an angle bisector, (b) a
perpendicular bisector, (c) an altitude, or (d) a median. List all that apply.
1. _____
2.
1.
3.
2. _____
3. _____
5.
4.
6.
4. _____
5. _____
6. _____
7.
8.
9.
7. _____
8. _____
9. _____
Identify each of the following as (a) an angle bisector, (b) a perpendicular bisector,
(c) an altitude, or (d) a median.
10. _____
10.
11.
11. _____
12. _____
13. _____
12.
13.
110
Section 6.2-Bisectors of Triangles
111
The perpendicular bisectors of ! ABC intersect at point G, or the angle bisectors of
intersect at point P. Find the indicated measure. Tell which theorem you used.
1. BG
4. PC
2. CG
!
XYZ
3. PS
5. AP
6. MP
Find the indicated measure. Tell which theorem you used
6. PJ  4x  8, PL  x  7
7. PN  6x  2, PM  8 x  14
Find PK .
Find PL.
5. LG  6x  14, NG   3x  22
Find MG and NG.
6. GL  4 x  2, GE  3x  2, GK  2 x  8
Find GJ and GE.
112
Section 6.3-Bisectors of Triangles
1. BL  12
Find BP and PL.
2. CP  16
Find PL and CL.
113
Point of Concurrency
Formed by
Sketch
Incenter
Circumcenter
Centroid
Orthocenter
Identify the correct point of concurrency:
1)
3)
2)
4)
114
Section 6.4-Midsegment Theorem
Find the values of the variables. Use formula Base = 2(Midsegment) or B = 2(M)
1. x = _____, y = _____, z = _____
x
2. x = _____
y
6
4
z
x
8
10
6
4
20
13
3. x = _____
4. x = _____
10
5
x
5. x = _____
x+1
6. x = _____
31
2x
x
5
60°
7. x = _____, y = _____
8. x = _____, y = _____
5x + 11
x
y
60
x + 23
3y - 9
40
2y + 6
115
Midsegments
Use formula Base = 2(Midsegment) or B = 2(M)
Find the values of the variables.
1. x = _____, y = _____, z = _____
9
2. x = _____
y
6
4
8
x
x
11
6
4
z
18
3. x = _____
4. x = _____
22
6
x
x+3
5. x = _____
6. x = _____, y = _____ z = _____
2x
40
24
x
z
y
60°
7. x = _____, y = _____, z = _____
8. x = _____, y = _____
3x + 12
z⁰
x
y
50
3y + 6
x+7
45
4y - 5
116
Section 6.5-Indirect Proof and Inequalities in One Triangle
**Complete Student Journal Pages 186-188**
In Exercises 1 and 2, list the angles of the given triangle from smallest to largest.
1.
2.
In Exercises 3 and 4, list the sides of the given triangle from shortest to longest.
3.
4.
In Exercises 5 and 6, is it possible to construct a triangle with the given side lengths? Explain.
5. 15, 37, 53
6. 9, 16, 8
7. Write an indirect proof that a triangle has at most one obtuse angle.
8. Describe the possible values of x in
the figure shown.
9. List the angles of the given triangle
from smallest to largest. Explain
your reasoning.
10. The shortest distance between two points is a straight line. Explain this statement in terms of the
Triangle Inequality Theorem (Theorem 6.11).
117
Name _________________________________________________________ Date _________
6.5
Practice B
In Exercises 1 and 2, list the angles of the given triangle from smallest to largest.
1.
2.
In Exercises 3 and 4, list the sides of the given triangle from shortest to longest.
3.
4.
5. Write an indirect proof that a right triangle has exactly two acute angles.
6. Is it possible to construct a triangle with side lengths 52 x  6, 3x  80, and
x2  41 if x  9? Explain.
7. The figure shows several triangles, with labeled side lengths. Which of the triangles are labeled
correctly? Explain.
8. Your friend claims that if you are given the three angle measures of a triangle,
you can construct a triangle that obeys the Triangle Inequality Theorem
(Theorem 6.11), even if you are not given any of the side lengths. Is your
friend correct? Explain your reasoning.
118
Section 6.6- Inequalities in Two Triangles
**Complete Student Journal Pages 191-192**
Copy and complete the statement with , , or  . Explain your reasoning.
1. AC _____ DF
4. KL _____ MN
7. m1 _____ m2
2. mHGI _____ mIGJ
5. BC _____ DE
3. m1 _____ m2
6. JI _____ GH
8. mU _____ mR
119
Write and solve an inequality for the possible values of x.
9.
11.
10.
12.
13.
Two sailboats started at the same location. Sailboat A traveled 5 miles west, then turned
29 toward the north and continued for 8 miles. Sailboat B first went south for 8 miles,
then turned 51 toward the east and continued for 5 miles. Which sailboat was farther
from the starting point? Explain your reasoning.
14.
How are the Hinge Theorem (Theorem 6.12) and the SAS Congruence Theorem
(Theorem 5.5) similar? How are they different? Explain your reasoning.
120