Name Class Date Extra Practice
Chapter 5
Lesson 5-1
Find the value of x.
1. 7x21
2. 25
7
3.
32
48
7
14
x
3x
48
In the figure at the right, points Q, R, and S are midpoints of kLMN .
4. Identify all pairs of parallel segments.
L
QR 6 MN, QS 6 LN, RS 6 LM
Q
5. If LR 5 25, find QS. 25
6. If MN 5 40, find QR. 20
R
M
7. If m/LNM 5 53, find m/SQM . 37
8. A sinkhole caused the sudden collapse of a large section of
highway. Highway safety investigators paced out the triangle
shown in the figure to help them estimate the distance across
the sinkhole. What is the distance across the sinkhole? 140 ft
150
182
ft
S
150
N
ft
280 ft
ft
Lesson 5-2
182
ft
Use the figure at the right for Exercises 9–12.
14. Since X
is on the
E
F
bisector of
3y
y15
10. Find the length of AD. 7
/BCN and
G
A
C
the bisector
5
11. Find the value of y. 2
2x23
x12
of /CBM, X
D
is equidistant
15
12. Find the length of EG. 2
from the
)
sides BM ,
13. Given: AP > AQ, BP > BQ, CP > CQ 14. Given: CX bisects /BCN .
) )
BC , CN
Prove: A, B, and C are collinear.
BX bisects /CBM . (/ Bisector
Prove: X is on the bisector Thm.).
It is given that
Therefore
P
of /A
each of A, B, and
X is
C is equidistant
M
equidistant
)
C
B
A from P and Q. By
from AM
X
B
the Conv. of the '
(containing
Bisector Thm., A,
)
)
BM ) and AN
B,
and
C
are
on
Q
(containing
the same line,
)
A
namely the '
CN ), the
C N
bisector of PQ.
sides of /A. By the Conv. of the / Bisector
Thm., X is on the bisector of /A.
9. Find the value of x.
B
5
)
)
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15
Name Class Date Extra Practice (continued)
Chapter 5
15. Find an equation in slope-intercept form for the perpendicular bisector of the
61
segment with endpoints H(23, 2) and K(7, 25). y 5 10
7 x 2 14
Lesson 5-3
Find the center of the circle that you can circumscribe about kABC.
16. A(2, 8) B(0, 8)
C(2, 2) (1, 5)
17. A(23, 6) B(23, 22)
C(7, 6) (2, 2)
18. A(4, 3) B(24, 23)
C(4, 23) (0, 0)
Use the figure at the right for Exercises 19 and 20.
R
19. What is the point of concurrency of the angle bisectors
of nQRS? C
Y
X A
B
20. Find the value of x if QC 5 2x 1 3 and
S
Z
Q
CS 5 5 2 4(x 1 1). 21
3
C
21. Find the center of the circle that you can circumscribe about
the triangle with vertices A(1, 3), B(5, 8), and C (6, 3). (3.5, 5.1)
22. Draw an acute triangle and construct its inscribed circle.
Check students’ work. The diagram should show three acute '. The
construction should include bisectors of at least two of the '. The final
diagram should show an incircle (tangent to the three sides of the n).
Lesson 5-4
In kDEF , L is the centroid.
E
23. If HL 5 30, find LF and HF. 60, 90
H
J
24. If KE 5 15, find KL and LE. 5, 10
D
25. If DL 5 24, find LJ and DJ. 12, 36
L
F
K
Is AB a median, an altitude, a perpendicular bisector, or none of these?
How do you know?
A
26. 27. 28.
A
B
A
B
B
Median; the line intersects
A and bisects a side at B.
Altitude; the segment intersects
A and is ' to a side at B.
' bisector; the segment is
perpendicular to and bisects
a side at B, but it does not
intersect a vertex.
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16
Name Class Date Extra Practice (continued)
Chapter 5
Find the coordinates of the orthocenter of kABC.
29. A(21, 0)
30. A(2, 5)
31. A(21, 21)
B(0, 4) B(6, 2) B(4, 0)
7
C(3, 0)
C(2, 21)
C(2, 4) a 25
a0, 34 b
a 17
4 , 2b
11 , 11 b
32. Tell which line contains each point for nABC.
a. the circumcenter line k
b. the orthocenter
B
line m
c. the centroid
line <
d. the incenter
line n
m
n
k
A
C
Lesson 5-5
Write the first step of an indirect proof.
33. nABC is a right triangle. Assume nABC is not a right n .
34. Points J, K, and L are collinear. Assume points J, K, and L are not collinear.
35. Lines / and m are not parallel. Assume lines < and m are n.
36. ~XYZV is a square. Assume ~XYZV is not a square.
Identify the two statements that contradict each other.
37. I. The perimeter of nXYZ is 10 cm.
38. I. nQRS is isoceles.
II. No side of nXYZ is longer than 5 cm.
II. nQRS has at least one acute angle.
III. No side of nXYZ is shorter than 4 cm.
III. nQRS is equiangular.
I and III
I and III
39. Suppose you know that /A is an obtuse angle in nABC. You want to prove
that /B is an acute angle. What assumption would you make to give an
indirect proof? /B is not an acute angle.
Lesson 5-6
Can a triangle have sides with the given lengths? Explain.
40. 2 in., 3 in., 5 in.
No; 2 1 3 6 5.
41. 9 cm, 11 cm, 15 cm
Yes; 9 1 11 . 15.
42. 8 ft, 9 ft, 18 ft No; 8 1 9 6 18.
List the sides of each triangle in order from shortest to longest.
43. N
R
82° 44°
44. P
RN, RS, NS
S
B
53°
60°
Q
45.
JB, PB, PJ
J
M
46°
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17
D
MQ, QD, MD
Name Class Date Extra Practice (continued)
Chapter 5
46. In nPQR, m/P 5 55, m/Q 5 82, and m/R 5 43. List the sides of the
triangle in order from shortest to longest. PQ, QR, PR
47. In nMNS, MN 5 7, NS 5 5, and MS 5 9. List the angles of the triangle in
order from smallest to largest. /M, /S, /N
48. Two sides of a triangle have side lengths 8 units and 17 units. Describe the
lengths x that are possible for the third side. 9 , x , 25
Lesson 5-7
Write an inequality relating the given side lengths. If there is not enough
information to reach a conclusion, write no conclusion.
49. AC and XZ
A
50. DF and FG
AC , XZ
X
101°
Y
D
FG . DF
E
51°
G
44°
C
75°
Z
B
F
Find the range of possible values for each variable.
51. 18
17
13
13
12
52.
11 , x , 29
6
45°
17
5x210°
9
18
3x27°
18
30°
7
3
15
, x , 37
3
Copy and complete with S or R. Explain your reasoning.
53. m/AEB
m/EBD
C
, by Converse of the Hinge Theorem A
21
B
79°
54. AB
BC
. by the Hinge Theorem
55. BC
ED
, by the Hinge Theorem (AB . BC, ED . AB,
so ED . BC)
E
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18
59°
23
D