Download Activity 32 Practice KEY - Newell-Math

More About Quadrilaterals
A 4-gon Conclusion
ACTIVITY 32
continued
6. Which of the following additional pieces of
information would allow you to prove that
ABCD is a parallelogram?
ACTIVITY 32 PRACTICE
Answer each item. Show your work.
Lesson 32-1
1. Given quad RSTU with coordinates R(0, 0),
S(−2, 2), T(6, 6), and U(8, 4).
a. Show that quad RSTU is a parallelogram by
finding the slope of each side.
b. Show that quad RSTU is a parallelogram by
finding the length of each side.
c. Show that quad RSTU is a parallelogram by
showing that the diagonals bisect each other.
2. Write a proof using the theorem in Item 2 of
Lesson 32-1 as the last reason.
Given: ABC ≅ FED
CD ≅ CG
A
B
E
D
C
A. AD || BC
B. AD ≅ BC
C. AB || DC
D. AB ≅ DC
Lesson 32-2
7. Each of the following sets of given information is
sufficient to prove that SPAR is a rectangle
except:
S
P
Prove: ACDF
A
K
C
F
G
D
E
3. Write a proof using the theorem in Item 4 of
Lesson 32-1 as the last reason.
© 2017 College Board. All rights reserved.
© 2017 College Board. All rights reserved.
Given: JKLM
Y is midpt of ML.
9. Write an indirect proof.
Prove: JXLY
X
M
A
A. SPAR and ∠SPA ≅ ∠PAR
B. SK = KA = RK = KP
C. SPAR and ∠SKP ≅ ∠PKA
D. ∠RSP ≅ ∠SPA ≅ ∠PAR ≅ ∠ARS
8. Given FOUR with coordinates F(0, 6), O(10, 8),
U(13, 3), and R(3, 0). Show that FOUR is not a
rectangle.
X is midpt of JK.
J
R
Given: CE ≠ DF
K
Y
Prove: CDEF is not a rectangle.
C
D
F
E
L
4. Which of the following is not a sufficient
condition to prove a quadrilateral is a
parallelogram?
A. The diagonals bisect each other.
B. One pair of opposite sides are parallel.
C. Both pairs of opposite sides are congruent.
D. Both pairs of opposite angles are congruent.
5. Show that the quadrilateral with vertices (−2, 3),
(−2, −1), (1, 1), and (1, 5) is a parallelogram.
6. A
7. C
8. Method 1: Opposite sides are not congruent:
OF = 104 and RU = 109 or FR = 45 and
OU = 34.
Method 2: Consecutive sides are not perpendicular:
slope of RU = 3 ; slope of OU = −5 .
10
3
Method 3: Diagonals are not congruent:
UF = 178 and OR = 113.
10. What is the best name for a quadrilateral if the
diagonals are congruent and bisect each other?
A. parallelogram
B. rectangle
C. kite
D. trapezoid
11. Three vertices of a rectangle are (−4, −3), (8, 3),
and (5, 6). Show that the diagonals are congruent.
Activity 32 • More About Quadrilaterals
9.
Statements
1. CDEF is a
rectangle.
2. CE = DF
3. CE ≠ DF
4. CDEF is not
a rectangle.
10. B
11. See next page.
ACTIVITY PRACTICE
1. a. slope of RS = slope of TU = −1;
slope of ST = slope of RU = 1
2
b. RS = TU = 8; ST = RU = 80
c. midpoint of SU = midpoint of
RT = (3, 3)
2.
Statements
Reasons
1. ABC ≅ FED 1. Given
2. CPCTC
2. AC ≅ FD
3. CD ≅ CG
3. Given
4. CG ≅ AF
4. Given
5. CD ≅ AF
5. Transitive Property
6. ACDF
6. If both pairs of opp.
sides of a quad. are
≅, the quad. is a
parallelogram.
3.
Statements
1. JKLM
2. JK ML
(and JX YL)
3. JK = ML
CG ≅ AF
B
ACTIVITY 32 Continued
561
Reasons
1. Assumption
2. Diags. of a rect. are ≅.
3. Given
4. The assumption led to a
contradiction between
steps 2 and 3.
Reasons
1. Given
2. Def. of
parallelogram
3. Opposite sides of a
para. are congruent.
4. Mult. Property of
4. 1 JK = 1 ML
2
2
Equality
5. X is midpt of JK 5. Given
and Y is midpt
of ML
6. JX = 1 JK and 6. Def of midpoint
2
YL = 1 ML
2
7. JX = YL
7. Substitution
8. JXLY
8. If one pair of opp.
sides of a quad. are
≅ and , the quad.
is a parallelogram.
4. B
5. Length of the segment from
(−2, 3) to (1, 5):
[1 − (−2)]2 + (5 − 3)2 = 13
Length of segment from
(−2, −1) to (1, 1):
[1 − (−2)]2 + [1 − (−1)]2 = 13
Length of segment from
(−2, 3) to (−2, −1):
[−2 − (−2)]2 + (−1 − 3)]2 = 4
Length of segment from
(1, 1) to (1, 5):
(1 − 1)2 + (5 − 1)2 = 4
Slope of segment from
(−2, 3) to (1, 5): 5 − 3 = 2
1 − (−2) 3
Slope of the segment from (−2, −1)
1 − (−1) 2
to (1, 1):
=
1 − (−2) 3
Slope of segment from (−2, 3) to
(−2, −1): −1 − 3 = undefined
−2 − (−2)
Slope of segment from (1, 1) to (1, 5):
5 − 1 = undefined
1−1
Opposite sides are ≅ and ||, so the
figure is a parallelogram.
Activity 32 • More About Quadrilaterals
561
14.
Statements
1. NTI ≅ NTH
2. ∠NTI ≅ ∠NTH
3. ∠NTI and ∠NTH
are right angles.
4. NG ⊥ HI
5. NIGH
6. NIGH is a
rhombus.
15.
Statements
1. PE ≅
⊥ EA
2. PEAR
3. PEAR is a
rhombus.
4. PE ⊥ EA
5. ∠PEA is a right
angle.
6. PEAR is a
square.
More About Quadrilaterals
A 4-gon Conclusion
ACTIVITY 32
continued
Lesson 32-3
Lesson 32-4
12. Each of the following sets of given information is
sufficient to prove that HOPE is a rhombus
except:
15. Write a proof.
H
Reasons
1. Given
2. CPCTC
3. If two ≅ ∠s form
a linear pair,
each is a rt ∠.
4. Def. of ⊥
5. Given
6. If the diags. of a
para. are ⊥, the
para. is a
rhombus.
Reasons
1. Given
2. Given
3. If a para. has 2
consecutive ≅
sides, it is a
rhombus.
4. Given
5. Def. of
perpendicular
6. A rhombus with
a right angle is a
square.
Given: PEAR, PE ⊥ EA;
PE ≅ AE
O
Prove: PEAR is a square.
P
X
E
P
A. HX = XP = XE = XO
B. OH = OP = PE = HE
C. HOPE and ∠HXO ≅ ∠OXP
D. HOPE and HE = PE
13. Given DRUM with coordinates D(−2, −2),
R(−3, 3), U(2, 5), and M(3, 0). Show that
DRUM is not a rhombus.
14. Write a proof using the theorem in Item 5 of
Lesson 32-3 as the last reason.
Given: NIGH
NTI ≅ NTH
L
R
A
16. Given quad SOPH with coordinates S(−8, 0),
O(0, 6), P(10, 6), and H(2, 0). What is the best
name for this quadrilateral?
A. parallelogram
B. rectangle
C. rhombus
D. square
17. What is the best name for an equilateral
quadrilateral whose diagonals are congruent?
A. parallelogram
B. rectangle
C. rhombus
D. square
18. Rhombus JKLM has vertices J(0, 1) and K(3, 5).
a. What could be the coordinates of vertices L
and M? Justify your answer algebraically.
b. Is there more than one possible answer to part a?
Explain your reasoning.
Prove: NIGH is a rhombus.
N
E
I
T
MATHEMATICAL PRACTICES
H
G
Look For and Make Use of Structure
19. Why is every rhombus a parallelogram but not
every parallelogram a rhombus? Why is every
square a rectangle but not every rectangle a
square? Why is every square a rhombus but not
every rhombus a square?
C
D
See below.
Sample answer: Every rhombus is a
parallelogram because every
rhombus is a quadrilateral with both
pairs of opposite sides parallel, which
is the definition of a parallelogram;
not every parallelogram is a rhombus
because not every parallelogram has
four congruent sides. Every square is
a rectangle because every square is a
parallelogram with four right angles,
which is the definition of a rectangle;
not every rectangle is a square
®
because not every rectangle has four
562TheSpringBoard
Mathematics
6 • Triangles
and
Quadrilaterals
18.I, Unit
a. Sample
answer:
The
coordinates could be
11.
fourth vertexIntegrated
is (−1, −6).
The
congruent sides. Every square is a
L(8, 5) and M(5, 1).
length of the diagonal, which is the
rhombus because every square is a
Use the Distance Formula to find JK:
segment connecting
parallelogram with four congruent
2
2
(−4, −3) and (8, 3) is
JK = (3 − 0 ) + (5 − 1) = 25 = 5
sides, which is the definition of a
2
2
(8 − (−4)) + (3 − (−3)) = 144 + 36 = 180
rhombus; not every rhombus is a
For JKLM to be a rhombus, all side lengths
2
2
square because not(8every
rhombus
− (−4)) + (3 − (−3)) = 144 + 36 = 180 . The length of
must be 5 units, so KL = |8 − 3| = 5 units
has four right angles.
the diagonal, which is the segment
and JM = |5 − 0| = 5.
connecting (−1, −6) and (5, 6) is
2
2
ADDITIONAL PRACTICE
ML = (8 − 5) + (5 − 1) = 25 = 5
If students need more practice on the
(5 − (−1))2 + (6 − (−6))2 = 36 + 144 = 180
b. Yes; There are many possible answers; as long
concepts in this activity, see the 2
(5 − (−1)) + (6 − (−6))2 = 36 + 144 = 180. The lengths of
as vertices are chosen so that all 5 sides are
Additional Unit Practice in Teacher
the diagonals are equal.
congruent, JKLM is a rhombus.
Resources on SpringBoard Digital for
additional practice problems.
16.
17.
18.
19.
562
SpringBoard® Integrated Mathematics I, Unit 6 • Triangles and Quadrilaterals
© 2017 College Board. All rights reserved.
12. A
13. Method: 1 All sides are not
congruent: DR = UM = 26 and
RU = DM = 29
Method 2: Diagonals are not
perpendicular: slope of DU = 7
4
and slope of RM = − 1
2
© 2017 College Board. All rights reserved.
ACTIVITY 32 Continued