Download State whether each sentence is true or false. If false, replace the

Study Guide and Review
State whether each sentence is true or false. If false, replace the underlined word or phrase to make a
true sentence.
1. No angles in an isosceles trapezoid are congruent.
SOLUTION: By definition, an isosceles trapezoid is a trapezoid in which the legs are congruent, both pairs of base angles are
congruent, and the diagonals are congruent.
false, both pairs of base angles
ANSWER: false, both pairs of base angles
2. If a parallelogram is a rectangle, then the diagonals are congruent.
SOLUTION: A rectangle is a parallelogram with four right angles, opposite sides parallel, opposite sides congruent, opposite angles
congruent, consecutive angles are supplementary, and the diagonals bisect each other. The statement is true.
ANSWER: true
3. A midsegment of a trapezoid is a segment that connects any two nonconsecutive vertices.
SOLUTION: The midsegment of a trapezoid is the segment that connects the midpoints of the legs of the trapezoid. A diagonal is
a segment that connects any two nonconsecutive vertices.
false, diagonal
ANSWER: false, diagonal
4. The base of a trapezoid is one of the parallel sides.
SOLUTION: One of the parallel sides of a trapezoid is its base. The statement is true.
ANSWER: true
5. The diagonals of a rhombus are perpendicular.
SOLUTION: A rhombus has perpendicular diagonals. The statement is true.
ANSWER: true
6. The diagonal of a trapezoid is the segment that connects the midpoints of the legs.
SOLUTION: A diagonal is a segment that connects any two nonconsecutive vertices. The midsegment of a trapezoid is the
segment that connects the midpoint of the legs of the trapezoid.
false, midsegment
ANSWER: false, midsegment
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7. A rectangle is not always a parallelogram.
Page 1
SOLUTION: A rhombus has perpendicular diagonals. The statement is true.
ANSWER: Study
Guide and Review
true
6. The diagonal of a trapezoid is the segment that connects the midpoints of the legs.
SOLUTION: A diagonal is a segment that connects any two nonconsecutive vertices. The midsegment of a trapezoid is the
segment that connects the midpoint of the legs of the trapezoid.
false, midsegment
ANSWER: false, midsegment
7. A rectangle is not always a parallelogram.
SOLUTION: By definition a rectangle is a parallelogram with four right angles.
false, is always
ANSWER: false, is always
8. A quadrilateral with only one set of parallel sides is a parallelogram.
SOLUTION: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A trapezoid is a quadrilateral with exactly
one pair of parallel sides.
false, trapezoid
ANSWER: false, trapezoid
9. A rectangle that is also a rhombus is a square.
SOLUTION: By definition, a square is a parallelogram with four congruent sides and four right angles. The statement is true.
ANSWER: true
10. The leg of a trapezoid is one of the parallel sides.
SOLUTION: By definition, a trapezoid is a quadrilateral with exactly one pair of parallel sides. The parallel sides are called bases.
The nonparallel sides are called legs.
false, nonparallel
ANSWER: false, nonparallel
Find the sum of the measures of the interior angles of each convex polygon.
11. decagon
SOLUTION: A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle
measures.
Substitute n = 10 in
.
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The nonparallel sides are called legs.
false, nonparallel
ANSWER: Study
Guide and Review
false, nonparallel
Find the sum of the measures of the interior angles of each convex polygon.
11. decagon
SOLUTION: A decagon has ten sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle
measures.
Substitute n = 10 in
.
ANSWER: 1,440
12. 15-gon
SOLUTION: A 15-gon has fifteen sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle
measures.
Substitute n = 15 in
.
ANSWER: 2,340
13. SNOWFLAKES The snowflake decoration at the right is a regular hexagon. Find the sum of the measures of the
interior angles of the hexagon.
SOLUTION: A hexagon has six sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle
measures.
Substitute n = 6 in
.
ANSWER: 720
The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.
14. 135
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SOLUTION: Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the
Page 3
ANSWER: Study
Guide and Review
720
The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.
14. 135
SOLUTION: Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the
interior angle measures is 135n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle
.
measures can also be expressed as
ANSWER: 8
15. ≈ 166.15
SOLUTION: Let n = the number of sides in the polygon. Since all angles of a regular polygon are congruent, the sum of the
interior angle measures is about 166.15n. By the Polygon Interior Angles Sum Theorem, the sum of the interior angle
.
measures can also be expressed as
ANSWER: 26
Use
ABCD to find each measure.
16. SOLUTION: We know that consecutive angles in a parallelogram are supplementary.
So,
Substitute.
ANSWER: 65°
17. AD
SOLUTION: eSolutions
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We know that opposite sides of a parallelogram are congruent.
So, Page 4
ANSWER: Study
65°Guide and Review
17. AD
SOLUTION: We know that opposite sides of a parallelogram are congruent.
So, ANSWER: 18
18. AB
SOLUTION: We know that opposite sides of a parallelogram are congruent.
So, ANSWER: 12
19. SOLUTION: We know that opposite angles of a parallelogram are congruent.
So, ANSWER: 115°
ALGEBRA Find the value of each variable in each parallelogram.
20. SOLUTION: Since the opposite sides of a parallelogram are congruent, 3x – 6 = x + 4.
Solve for x.
3x – 6 = x + 4 Opp. sides of a parallelogram are .
2x – 6 = 4 Subtract x from each side.
2x = 10 Add 6 to each side.
x = 5 Divide each side by 2.
Since alternate interior angles are congruent,
.
5y = 60
y = 12
So, x = 5 and y = 12.
ANSWER: x = 5, y = 12
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So, x = 5 and y = 12.
ANSWER: Study
x =Guide
5, y =and
12 Review
21. SOLUTION: Since the opposite sides are congruent, 3y + 13 = 2y + 19.
Solve for y.
3y + 13 = 2y + 19
y =6
Since the opposite angles are congruent, 2x + 41 = 115.
Solve for x.
2x + 41 = 115
2x = 74
x = 37
ANSWER: x = 37, y = 6
22. DESIGN What type of information is needed to determine whether the shapes that make up the stained glass
window below are parallelograms?
SOLUTION: Sample answer: Review the definition of and theorems about parallelograms. A quadrilateral is a parallelogram if
both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and parallel or if both
pairs of opposite sides are parallel..The shapes can also be parallelograms if both pairs of opposite angles are
congruent or if the diagonals bisect each other.
ANSWER: Sample answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and
parallel, then the shapes are parallelograms. The shapes can also be parallelograms if both pairs of opposite angles
are congruent or if the diagonals bisect each other.
Determine whether each quadrilateral is a parallelogram. Justify your answer.
23. SOLUTION: The diagonals of the figure bisect each other. By Theorem 6.11 if the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a
parallelogram.
ANSWER: eSolutions
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6.11 by Cognero
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ANSWER: Sample answer: If both pairs of opposite sides are the same length or if one pair of opposite sides is congruent and
parallel,
shapes are parallelograms. The shapes can also be parallelograms if both pairs of opposite angles
Study
Guidethen
andthe
Review
are congruent or if the diagonals bisect each other.
Determine whether each quadrilateral is a parallelogram. Justify your answer.
23. SOLUTION: The diagonals of the figure bisect each other. By Theorem 6.11 if the diagonals of a quadrilateral bisect each other,
then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a
parallelogram.
ANSWER: yes, Theorem 6.11
24. SOLUTION: One pair of opposite sides are parallel and congruent. By Theorem 6.12 if one pair of opposite sides of a
quadrilateral is both parallel and congruent, then the quadrilateral is a parallelogram. No other information is needed
to determine that the figure is a parallelogram.
ANSWER: yes, Theorem 6.12
25. PROOF Write a two-column proof.
Given:
Prove: Quadrilateral EBFD is a parallelogram.
SOLUTION: You need to walk through the proof step by step. Look over what you are given and what you need to prove. Here,
you are given
. You need to prove that EBFD is a parallelogram. Use the properties that you
have learned about parallelograms to walk through the proof.
Given:
Prove: Quadrilateral EBFD is a parallelogram.
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1. ABCD,
(Given)
2. AE = CF (Def. of segs)
Page 7
you are given
. You need to prove that EBFD is a parallelogram. Use the properties that you
have learned about parallelograms to walk through the proof.
Study Guide and Review
Given:
Prove: Quadrilateral EBFD is a parallelogram.
1. ABCD,
(Given)
2. AE = CF (Def. of segs)
3.
(Opp. sides of a
)
4. BC = AD (Def. of segs)
5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.)
6. BF + CF = AE + ED (Subst.)
7. BF + AE = AE + ED (Subst.)
8. BF = ED (Subt. Prop.)
9.
(Def. of segs)
10.
(Def. of )
11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a
parallelogram.)
ANSWER: Given:
Prove: Quadrilateral EBFD is a parallelogram.
1. ABCD,
(Given)
2. AE = CF (Def. of segs)
3.
(Opp. sides of a
)
4. BC = AD (Def. of segs)
5. BC = BF + CF, AD = AE +ED (Seg. Add. Post.)
6. BF + CF = AE + ED (Subst.)
7. BF + AE = AE + ED (Subst.)
8. BF = ED (Subt. Prop.)
9.
(Def. of segs)
10.
(Def. of )
11. Quadrilateral EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a
parallelogram.)
ALGEBRA Find x and y so that the quadrilateral is a parallelogram.
26. eSolutions
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SOLUTION: We know that opposite angles of a parallelogram are congruent.
Page 8
9.
(Def. of segs)
10.
(Def. of )
11.Guide
Quadrilateral
EBFD is a parallelogram. (If one pair of opposite sides is parallel and congruent then it is a
Study
and Review
parallelogram.)
ALGEBRA Find x and y so that the quadrilateral is a parallelogram.
26. SOLUTION: We know that opposite angles of a parallelogram are congruent.
So, 12x + 72 = 25x + 20 and 3y + 36 = 9y - 12.
Solve for x.
12x + 72 = 25x + 20
72 = 13x + 20
52 = 13x
4 = x
Solve for y.
3y + 36 = 9y - 12
36 = 6y - 12
48 = 6y
8 = y
When x = 4 and y = 8 the quadrilateral is a parallelogram.
ANSWER: x = 4, y = 8
27. SOLUTION: We know that diagonals of a parallelogram bisect each other.
So,
.
Solve for x.
Alternate interior angles in a parallelogram are congruent.
Solve for y.
5y = 60
So, y = 12.
When x = 5 and y = 12 the quadrilateral is a parallelogram.
ANSWER: x = 5, y = 12
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28. PARKING The lines of the parking space shown below are parallel. How wide is the space (in inches)?
Page 9
When x = 4 and y = 8 the quadrilateral is a parallelogram.
ANSWER: Study
x =Guide
4, y =and
8 Review
27. SOLUTION: We know that diagonals of a parallelogram bisect each other.
So,
.
Solve for x.
Alternate interior angles in a parallelogram are congruent.
Solve for y.
5y = 60
So, y = 12.
When x = 5 and y = 12 the quadrilateral is a parallelogram.
ANSWER: x = 5, y = 12
28. PARKING The lines of the parking space shown below are parallel. How wide is the space (in inches)?
SOLUTION: Since the distance between two parallel lines is the same, we can write the equation 6x + 12 = 5x + 20 and then
solve for x.
6x + 12 = 5x + 20
x + 12 = 20
x = 8
Substitute x = 8 in 5x + 20.
5x + 20 = 5(8) + 20
= 60
So, the length of the space is 60 inches.
ANSWER: 60 in.
ALGEBRA Quadrilateral EFGH is a rectangle.
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Page 10
So, the length of the space is 60 inches.
ANSWER: Study
Guide and Review
60 in.
ALGEBRA Quadrilateral EFGH is a rectangle.
29. If , find
.
SOLUTION: All four angles of a rectangle are right angles. So,
Substitute.
ANSWER: 33
30. If , find
.
SOLUTION: All four angles of a rectangle are right angles. So,
Substitute.
ANSWER: 77
31. If FK = 32 feet, find EG.
SOLUTION: We know that diagonals of a rectangle are congruent and bisect each other. So, EG = FH, FK = KH, and EK = KG.
FH = FK + KH Diagonals of a rectangle bisect each other.
= FK + FK FK = KH, substitution
= 32 + 32 Substitute.
= 64 Add.
EG is the same length as FH so EG = 64 feet.
ANSWER: 64
32. Find
SOLUTION: All four angles of a rectangle are right angles. So,
ANSWER: 180
33. If EF = 4x – 6 and HG = x + 3, find EF.
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SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, EF = HG.
Page 11
All four angles of a rectangle are right angles. So,
ANSWER: Study
Guide and Review
180
33. If EF = 4x – 6 and HG = x + 3, find EF.
SOLUTION: The opposite sides of a rectangle are parallel and congruent. Therefore, EF = HG.
EF = HG Opp. sides of rectangle are congruent.
4x – 6 = x + 3 Substitution.
3x – 6 = 3 Subtract x from each side.
3x = 9 Add 6 to each side.
x = 3 Divide each side by 3.
Substitute x = 3 into 4x - 6 to find EF.
EF = 4x – 6 Original equation.
= 4(3) – 6 x = 3
= 12 – 6 Multiply.
= 6 Subtract.
So, EF = 6.
ANSWER: 6
ALGEBRA ABCD is a rhombus. If EB = 9, AB = 12 and
, find each measure.
34. AE
SOLUTION: The diagonals of a rhombus are perpendicular. So, use the Pythagorean Theorem.
Since the length must be positive, AE = 7.9.
ANSWER: 7.9
35. SOLUTION: All the four sides of a rhombus are congruent. So,
is an isosceles triangle. Therefore, ANSWER: eSolutions
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36. CE
Page 12
Since the length must be positive, AE = 7.9.
ANSWER: Study
7.9Guide and Review
35. SOLUTION: All the four sides of a rhombus are congruent. So,
is an isosceles triangle. Therefore, ANSWER: 55
36. CE
SOLUTION: The diagonals of a rhombus are perpendicular. Use AE to find CE.
Use the Pythagorean Theorem.
Since the length must be positive, AE = 7.9.
CE = AE = 7.9
ANSWER: 7.9
37. SOLUTION: The diagonals are perpendicular to each other. So, in the right triangle EAB,
All the four sides of a rhombus are congruent. So,
is an isosceles triangle. Therefore, ANSWER: 35
38. LOGOS A car company uses the symbol shown at the right for their logo. If the inside space of the logo is a
rhombus, what is the length of FJ?
SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ = 2.5 cm.
ANSWER: 2.5 cm
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COORDINATE GEOMETRY Given each set of vertices,
determine whether QRST is a rhombus, a rectangle, or a square. List all that apply. Explain.
Page 13
All the four sides of a rhombus are congruent. So,
is an isosceles triangle. Therefore, ANSWER: Study
Guide and Review
35
38. LOGOS A car company uses the symbol shown at the right for their logo. If the inside space of the logo is a
rhombus, what is the length of FJ?
SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ = 2.5 cm.
ANSWER: 2.5 cm
COORDINATE GEOMETRY Given each set of vertices,
determine whether QRST is a rhombus, a rectangle, or a square. List all that apply. Explain.
39. Q(12, 0), R(6, -6), S(0, 0), T(6, 6)
SOLUTION: First graph the quadrilateral.
Use the distance formula to find the length of each side of QRST.
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Study Guide and Review
So, all sides are congruent. The quadrilateral is a rhombus.
Check to see whether we can say more: are consecutive sides perpendicular?
Since the products of the slopes of consecutive sides are -1, the sides are perpendicular.
So, the quadrilateral is also a rectangle and a square.
ANSWER: Rectangle, rhombus, square; all sides are
, consecutive are
.
40. Q(–2, 4), R(5, 6), S(12, 4), T(5, 2)
SOLUTION: First graph the quadrilateral.
Use the distance formula to find the length of each side of QRST.
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Study Guide and Review
All the sides are congruent.
If the diagonals of the parallelogram are congruent, then it is a rectangle. Use the Distance Formula to find the
lengths of the diagonals.
, the diagonals are not congruent. So, QRST is not a rectangle. Since the figure is not a rectangle, it also
Since
cannot be a square.
Check whether the two diagonals are perpendicular.
Undefined slope and 0 slope are perpendicular, so the diagonals are perpendicular. It is a rhombus.
ANSWER: Rhombus; all sides are
, diagonals are
.
Find each measure.
41. GH
SOLUTION: Use the Pythagorean Theorem.
Since the length must be positive, GH = 19.2.
ANSWER: 19.2Manual - Powered by Cognero
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Undefined slope and 0 slope are perpendicular, so the diagonals are perpendicular. It is a rhombus.
ANSWER: Study
Guide and Review
Rhombus; all sides are
, diagonals are
.
Find each measure.
41. GH
SOLUTION: Use the Pythagorean Theorem.
Since the length must be positive, GH = 19.2.
ANSWER: 19.2
42. SOLUTION: The trapezoid WZXY is an isosceles trapezoid. So, each pair of base angles is congruent. So,
The sum of the measures of the angles of a quadrilateral is 360.
Let
.
So,
ANSWER: 68
43. DESIGN Renee designed the square tile as an art project.
a. Describe a way to determine if the trapezoids in the design are isosceles.
b. If the perimeter of the tile is 48 inches and the perimeter of the red square is 16 inches, what is the perimeter of
one of the trapezoids?
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So,
ANSWER: Study
Guide and Review
68
43. DESIGN Renee designed the square tile as an art project.
a. Describe a way to determine if the trapezoids in the design are isosceles.
b. If the perimeter of the tile is 48 inches and the perimeter of the red square is 16 inches, what is the perimeter of
one of the trapezoids?
SOLUTION: a. Sample answer: A trapezoid is isosceles if the legs are congruent. The legs of the trapezoids are part of the
diagonals of the square tile. The diagonals of a square bisect opposite angles, so each base angle of each trapezoid
measures 45°. One pair of sides is parallel and the base angles are congruent.
b. The perimeter of a square is given by 4s, where s is the side length. Solving 48 = 4s1 and 16 = 4s2, we find that
the tile is 12 in. long on a side and the red square is 4 in. long on a side. Now all that remains is to find the two other
sides of the trapezoid.
A diagonal of the tile is
. So the length of each nonparallel side of
in, and a diagonal of the red square is a trapezoid is
in.
Add to find the perimeter of the trapezoid.
ANSWER: a. Sample answer: The legs of the trapezoids are part of the diagonals of the square. The diagonals of a square
bisect opposite angles, so each base angle of a trapezoid measures 45°. One pair of sides is parallel and the base angles are congruent.
b.
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Practice Test - Chapter 6
Find the sum of the measures of the interior angles of each convex polygon.
1. hexagon
SOLUTION: A hexagon has six sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle
measures.
Substitute n = 6 in
.
ANSWER: 720
2. 16-gon
SOLUTION: A 16-gon has sixteen sides. Use the Polygon Interior Angles Sum Theorem to find the sum of its interior angle
measures.
Substitute n = 16 in
.
ANSWER: 2520
3. ART Jen is making a frame to stretch a canvas over for a painting. She nailed four pieces of wood together at what
she believes will be the four vertices of a square.
a. How can she be sure that the canvas will be a square?
b. If the canvas has the dimensions shown below, what are the missing measures?
SOLUTION: a. Sample answer: A quadrilateral is a square if it has diagonals that are congruent and perpendicular or all sides are
congruent with 4 right angles. She should measure the angles at the vertices to see if they are 90 or she can check to
see if the diagonals are congruent and perpendicular.
b. Each side of a square have the same measure so x = 2 ft. Each angle of a square is a right angle so y = 90.
ANSWER: a. Sample answer: She should measure the angles at the vertices to see if they are 90 or she can check to see if the
diagonals are congruent and perpendicular.
b. x = 2 ft, y = 90
Quadrilateral ABCD is an isosceles trapezoid.
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Page 1
ANSWER: a. Sample answer: She should measure the angles at the vertices to see if they are 90 or she can check to see if the
diagonals
congruent
Practice
Testare
- Chapter
6 and perpendicular.
b. x = 2 ft, y = 90
Quadrilateral ABCD is an isosceles trapezoid.
4. Which angle is congruent to
?
SOLUTION: The base angles of an isosceles triangle are congruent so
is congruent to
.
ANSWER: 5. Which side is parallel to
?
SOLUTION: The bases of a trapezoid are parallel so is parallel to
.
ANSWER: 6. Which segment is congruent to
?
SOLUTION: The diagonals of an isosceles trapezoid are congruent so
is congruent to
.
ANSWER: The measure of the interior angles of a regular polygon is given. Find the number of sides in the polygon.
7. 900
SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior
.
angle measures can also be expressed as
ANSWER: 7
8. 1980
SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior
.
angle measures can also be expressed as
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Page 2
ANSWER: Practice
Test - Chapter 6
7
8. 1980
SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior
.
angle measures can also be expressed as
ANSWER: 13
9. 2880
SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior
.
angle measures can also be expressed as
ANSWER: 18
10. 5400
SOLUTION: Let n be the number of sides in the polygon. By the Polygon Interior Angles Sum Theorem, the sum of the interior
.
angle measures can also be expressed as
ANSWER: 32
11. MULTIPLE CHOICE If QRST is a parallelogram, what is the value of x?
A 11 C 13
B 12 D 14
SOLUTION: Diagonals of a parallelogram bisect each other. So, 14x – 34 = 12x – 8.
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14x – 34 = 12x – 8 Diag. bisect each other.
2x – 34 = – 8 Subtract 12x from each side.
Page 3
ANSWER: Practice
Test - Chapter 6
32
11. MULTIPLE CHOICE If QRST is a parallelogram, what is the value of x?
A 11 C 13
B 12 D 14
SOLUTION: Diagonals of a parallelogram bisect each other. So, 14x – 34 = 12x – 8.
Solve for x.
14x – 34 = 12x – 8 Diag. bisect each other.
2x – 34 = – 8 Subtract 12x from each side.
2x = 26 Add 34 to each side.
x = 13 Divide each side by 2.
So, the correct option is C.
ANSWER: C
If CDFG is a kite, find each measure.
12. GF
SOLUTION: Use the Pythagorean Theorem.
Since the length must be positive, GF = 5.
ANSWER: 5
13. SOLUTION: eSolutions
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Since a kite can only have one pair of opposite congruent angles and
Let x be the measure of angle D.
Page 4
Since the length must be positive, GF = 5.
ANSWER: Practice
Test - Chapter 6
5
13. SOLUTION: Since a kite can only have one pair of opposite congruent angles and
Let x be the measure of angle D.
The sum of the measures of the angles of a quadrilateral is 360.
So,
ANSWER: 122
ALGEBRA Quadrilateral MNOP is a rhombus. Find each value or measure.
14. SOLUTION: Since the diagonals of a rhombus are perpendicular,
by the definition of perpendicular lines.
ANSWER: 90
15. If PR = 12, find RN.
SOLUTION: In a rhombus, diagonals bisect each other. So, RN = PN = 12.
ANSWER: 12
16. If , find
.
SOLUTION: Since MNOP is a rhombus, diagonal
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bisects . Therefore,
.
Page 5
SOLUTION: In a rhombus, diagonals bisect each other. So, RN = PN = 12.
ANSWER: Practice
Test - Chapter 6
12
16. If , find
.
SOLUTION: Since MNOP is a rhombus, diagonal
bisects . Therefore,
.
ANSWER: 62
17. CONSTRUCTION The Smiths are building an addition to their house. Mrs. Smith is cutting an opening for a new
window. If she measures to see that the opposite sides are congruent and that the diagonal measures are congruent,
can Mrs. Smith be sure that the window opening is rectangular? Explain.
SOLUTION: Sample answer: Yes, that is enough to show that the opening is a rectangle. Since each pair of opposite sides are the
same length, the opening is a parallelogram. By Theorem 6.14, if the diagonals of a parallelogram are congruent then
it is a rectangle. ANSWER: Sample answer: Yes. If it is a rectangle, the diagonals are congruent.
Use
JKLM to find each measure.
18. SOLUTION: Opposite angles of a parallelogram are congruent.
So,
ANSWER: 109
19. JK
SOLUTION: Opposite sides of a parallelogram are congruent.
So, JK = 6.
ANSWER: 6
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SOLUTION: Consecutive angles in a parallelogram are supplementary.
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Opposite sides of a parallelogram are congruent.
So, JK = 6.
ANSWER: Practice
Test - Chapter 6
6
20. SOLUTION: Consecutive angles in a parallelogram are supplementary.
So,
ANSWER: 71
ALGEBRA Quadrilateral DEFG is a rectangle.
21. If DF = 2(x + 5) – 7 and EG = 3(x – 2), find EG.
SOLUTION: The diagonals of a rectangle are congruent to each other. So, FD = EG.
Use the value of x to find EG.
EG = 3(9 – 2) = 21
ANSWER: 21
22. If , find
SOLUTION: .
.
Substitute
in .
ANSWER: 22
eSolutions
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= 14- +Powered
2x andbyGF
= 4(x
23. If DE
SOLUTION: – 3) + 6, find GF.
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ANSWER: Practice
Test - Chapter 6
22
23. If DE = 14 + 2x and GF = 4(x – 3) + 6, find GF.
SOLUTION: Opposite sides of a rectangle are congruent.
DE = GF
14 + 2x = 4( x – 3) + 6 Substitute.
14 + 2x = 4x – 12 + 6 Distributive Property
14 + 2x = 4x – 6 Simplify.
14 – 2x = –6 Subtract 4x from each side.
–2x = –20 Subtract 14 from each side.
2x = 20 Divide each side by -1.
x =10 Divide each side by 2.
Use the value of x to find GF.
GF = 4(x – 3) + 6 Original equation
= 4(10 - 3) + 6 Substitute.
= 4(7) + 6 Subtract.
= 34 Simplify.
ANSWER: 34
Determine whether each quadrilateral is a parallelogram. Justify your answer.
24. SOLUTION: Each pair of opposite angles are congruent. By Theorem 6.10 if both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram. No other information is needed to determine that the figure is a
parallelogram.
ANSWER: Yes, opposite angles are congruent.
25. SOLUTION: There are 2 pairs of consecutive angles that are congruent. Since opposite sides are not congruent, this fails
Theorem 6.9. If both pairs of opposite sides are congruent, the quadrilateral is a parallelogram. This is not a
parallelogram.
ANSWER: No, opposite sides are not congruent.
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