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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.
Thus, the statement "No angles in an isosceles trapezoid are congruent." is false. It should be "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.
Thus, the statement "A midsegment of a trapezoid is a segment that connects any two nonconsecutive vertices." is
false. It should be " 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.
Thus, the statement "The diagonal of a trapezoid is the segment that connects the midpoints of the legs." is false. It
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should be the " midsegment".
ANSWER: SOLUTION: A rhombus has perpendicular diagonals. The statement is true.
ANSWER: Study
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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.
Thus, the statement "The diagonal of a trapezoid is the segment that connects the midpoints of the legs." is false. It
should be the " midsegment".
ANSWER: false, midsegment
7. A rectangle is not always a parallelogram.
SOLUTION: By definition a rectangle is a parallelogram with four right angles.
Thus, the statement "A rectangle is not always a parallelogram." is false. It should be "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.
Thus, the statement "A quadrilateral with only one set of parallel sides is a parallelogram." is false. It should be
"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.
Thus, the statement "The leg of a trapezoid is one of the parallel sides." is false. It should be "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
= 10 in by Cognero .
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The nonparallel sides are called legs.
Thus, the statement "The leg of a trapezoid is one of the parallel sides." is false. It should be "nonparallel".
ANSWER: Study
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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
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ANSWER: Study
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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: We know that opposite sides of a parallelogram are congruent.
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ANSWER: Study
65Guide 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: 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.
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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
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Guidethen
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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)
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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.
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SOLUTION: We know that opposite angles of a parallelogram are congruent.
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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
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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)?
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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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29. If , find
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.
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.
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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
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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.
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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.
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,
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ANSWER: 68
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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.
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
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