Download Answers for the lesson “Use Properties of Parallelograms”

Answers for the lesson “Use Properties of
Parallelograms”
LESSON
8.2
15. x 5 4, y 5 4
Skill Practice
1. A parallelogram is a
16. A
quadrilateral with both pairs of
opposite sides parallel; opposite
sides are congruent, opposite
angles are congruent, consecutive
angles are supplementary, and the
diagonals bisect each other.
2. mŽB 5 1158 since consecutive
angles are supplementary and
mŽC 5 658 and mŽD 5 1158
since opposite angles are
congruent.
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
4. m 5 5, n 5 12
6. p 5 60
P
10. 858
1028
parallelogram are congruent.
19. ŽDAC; alternate interior angles
are congruent.
20. 478; opposite angles of a
parallelogram are congruent.
21. 478; consecutive angles of a
are congruent.
23. 1208; ŽEJF and ŽFJG are a
8. g 5 61, h 5 9
12.
18. ŽBCD; opposite angles of a
22. 868; alternate interior angles
7. d 5 126, z 5 28
9. 1298
parallelogram are congruent.
parallelogram are supplementary
and alternate interior angles are
congruent.
3. x 5 9, y 5 15
5. a 5 55
17. }
BC; opposite sides of a
linear pair.
11. 618
788
Q
24. 858; Alternate Interior Angles
Theorem with ŽHEG
25. 358; Triangle Sum Theorem
26. 45; Alternate Interior Angles
788
S
Theorem with ŽHGE
1028
R
mŽS 5 788, mŽP 5 1028,
mŽQ 5 788, mŽR 5 1028
13. a 5 3, b 5 10
14. m 5 4, n 5 3
27. 1308; sum of the measures of
ŽHGE and ŽEGF
28. 508; consecutive angles are
supplementary, ŽHGF and
ŽEHG.
Geometry
Answer Transparencies for Checking Homework
228
29. C
37. (22, 4), (4, 0), (8, 8);
30. 368, 1448
y
31. 268, 1548
32. ŽB and ŽA are consecutive
angles and thus are supplementary
which makes mŽA 5 1308.
D
C
33. 20, 608; UV 5 TS 5 QR using the
1
fact that opposite sides are
congruent and the Transitive
Property of Equality.
ŽTUS > ŽVSU by the
Alternate Interior Angles
Congruence Theorem and
mŽTSU 5 608 by the Triangle
Sum Theorem.
34.
M
4y 1 5
B
A
21
x
y
B
C
1
N
A
21
D
x
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
y
22x 1 37
Q
x25
y 1 14
B
P
C
52
35. Sample answer: In a
parallelogram, opposite angles
are congruent. ŽA and ŽC are
opposite angles but not congruent.
36. 168
D
1
21
A
x
In each quadrilateral each pair of
opposite sides is parallel.
Geometry
Answer Transparencies for Checking Homework
229
Problem Solving
corresponding angles are equal.
So the parallelograms are similar.
38. 1408; ŽC and ŽD are
consecutive angles and therefore
are supplementary.
42. Statements (Reasons)
1. ABCD is a parallelogram.
(Given)
39. a. 3 in.
b. 708
c. It decreases; it gets longer; the
3. ŽCBD > ŽADB,
ŽCDB > ŽADB
(Alternate Interior Angles
Congruence Theorem)
sum of the measures of the
interior angles always is 3608.
As mŽQ increases so does
mŽS therefore mŽP must
decrease to maintain the sum
of 3608. As mŽQ decreases
mŽP increases, moving Q
farther away from S.
4. }
BD > }
BD
(Reflexive Property of
Segment Congruence)
5 nABD > nCDB
40. 8
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
2. }
BD i }
AC, }
AB i }
CD
(Definition of a parallelogram)
(ASA)
41. Sample:
6. ŽA > ŽC, ŽB > ŽD
s are > .)
(Corr. parts of > n
E
D
D
B
C
A
B
C
A
F
G
B
C
A
H
J
Since nABC > nDCB you
know ŽACB > ŽDBC and
ŽABC > ŽDCB. By the
Alternate Interior Angles
Converse, }
BD i }
AC and }
AB i }
CD,
so ABDC is a parallelogram.
If 6 more triangles are positioned
as shown, the sides of the resulting figure are all twice the length
of the corresponding sides of
ABDC and the measures of
43. Sample answer: Given that PQRS
is a parallelogram you know that
}
QR i }
PS with }
QP being a
transversal. By definition and
the fact that they are consecutive
interior angles, ŽQ and ŽP
are supplementary using the
Consecutive Interior Angles
Theorem. So x8 1 y8 5 1808
by the definition of
supplementary angles.
Geometry
Answer Transparencies for Checking Homework
230
DA 1 DC
DE 1 DG
AC
AF
AC
DE 1 DG
}, which
5
implies }
AC
AF
DE 1 DG
5 1, which
implies }
AF
} 5 }, which
44. Statements (Reasons)
1. PQRS is a parallelogram.
(Given)
2. }
PQ > }
RS, }
QR > }
SP
(If a quadrilateral is a
parallelogram, then its
opposite sides are congruent.)
implies DE 1 DG 5 AF.
3. ŽQPR > ŽSRP,
ŽPQS > ŽRSQ,
Ž RPS > Ž QRP,
ŽPSQ > ŽRQS
(Alternate Interior Angles
Congruence Theorem)
Copyright © Houghton Mifflin Harcourt Publishing Company. All rights reserved.
4. nPMQ > nRMS,
nQMR > nSMP
(ASA)
5. }
QM > }
SM, }
PM > }
RM
s are > .)
(Corr. parts of > n
6. M bisects }
QS and }
PR.
(Definition of
segment bisector)
45. Sample answer: nDCG , nACF
and nDAE , nACF using the
AA Similarity Postulate.
DG
AF
DC
AC
DE
AF
DA
AC
} 5 } and } 5 } since the
ratio of corresponding sides of
similar triangles are equal.
Adding you get
DE
AF
DG
AF
DA
AC
DC
AC
} 1 } 5 } 1 }, which
implies
Geometry
Answer Transparencies for Checking Homework
231