Download So - cloudfront.net

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
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.
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
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.
5. The diagonals of a rhombus are perpendicular.
SOLUTION: A rhombus has perpendicular diagonals. The statement is 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
7. A rectangle is not always a parallelogram.
SOLUTION: By definition a rectangle is a parallelogram with four right angles.
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
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.
eSolutions Manual - Powered by Cognero
10. The leg of a trapezoid is one of the parallel sides.
SOLUTION: Page 1
SOLUTION: A parallelogram is a quadrilateral with both pairs of opposite sides parallel. A trapezoid is a quadrilateral with exactly
Study
Guide
Review
one
pair ofand
parallel
sides.
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.
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
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
.
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
.
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
.
eSolutions Manual - Powered by Cognero
Page 2
The measure of an interior angle of a regular polygon is given. Find the number of sides in the polygon.
14. 135
Substitute n = 15 in
.
Study Guide and Review
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
.
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
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
Use
ABCD to find each measure.
16. SOLUTION: We know that consecutive angles in a parallelogram are supplementary.
So,
Substitute.
eSolutions Manual - Powered by Cognero
Page 3
Study Guide and Review
Use
ABCD to find each measure.
16. SOLUTION: We know that consecutive angles in a parallelogram are supplementary.
So,
Substitute.
17. AD
SOLUTION: We know that opposite sides of a parallelogram are congruent.
So, 18. AB
SOLUTION: We know that opposite sides of a parallelogram are congruent.
So, 19. SOLUTION: We know that opposite angles of a parallelogram are congruent.
So, 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
eSolutions Manual - Powered by Cognero
So, x = 5 and y = 12.
.
Page 4
19. SOLUTION: WeGuide
knowand
that Review
opposite angles of a parallelogram are congruent.
Study
So, 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.
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
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 Page 5
eSolutions Manual - Powered by Cognero
congruent or if the diagonals bisect each other.
Determine whether each quadrilateral is a parallelogram. Justify your answer.
Since the opposite angles are congruent, 2x + 41 = 115.
Solve for x.
2x + 41 = 115
2xGuide
= 74 and Review
Study
x = 37
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.
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.
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.
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.
eSolutions Manual - Powered by Cognero
Page 6
SOLUTION: One pair of opposite sides are parallel and congruent. By Theorem 6.12 if one pair of opposite sides of a
quadrilateral
both parallel and congruent, then the quadrilateral is a parallelogram. No other information is needed
Study
Guide andisReview
to determine that the figure is a parallelogram.
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.
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. 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
+ 20by Cognero
72 = 13x
eSolutions
Manual - Powered
52 = 13x
4 = x
Page 7
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.
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.
28. PARKING The lines of the parking space shown below are parallel. How wide is the space (in inches)?
eSolutions Manual - Powered by Cognero
SOLUTION: Page 8
Solve for y.
5y = 60
So, y = 12.
Guide and Review
Study
When x = 5 and y = 12 the quadrilateral is a parallelogram.
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.
ALGEBRA Quadrilateral EFGH is a rectangle.
29. If , find
.
SOLUTION: All four angles of a rectangle are right angles. So,
Substitute.
30. If , find
.
SOLUTION: All four angles of a rectangle are right angles. So,
Substitute.
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.
eSolutions Manual - Powered by Cognero
= 64 Add.
EG is the same length as FH so EG = 64 feet.
Page 9
All four angles of a rectangle are right angles. So,
Substitute.
Study Guide and Review
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.
32. Find
SOLUTION: All four angles of a rectangle are right angles. So,
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.
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.
eSolutions Manual - Powered by Cognero
Since the length must be positive, AE = 7.9.
Page 10
= 4(3) – 6 x = 3
= 12 – 6 Multiply.
= 6 Subtract.
Guide and Review
Study
So, EF = 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.
35. SOLUTION: All the four sides of a rhombus are congruent. So,
is an isosceles triangle. Therefore, 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
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, 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?
eSolutions Manual - Powered by Cognero
Page 11
SOLUTION: The diagonals are perpendicular to each other. So, in the right triangle EAB,
AllGuide
the four
sides
of a rhombus are congruent. So,
Study
and
Review
is an isosceles triangle. Therefore, 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.
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.
eSolutions Manual - Powered by Cognero
So, all sides are congruent. The quadrilateral is a rhombus.
Check to see whether we can say more: are consecutive sides perpendicular?
Page 12
SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ = 2.5 cm.
Study Guide and Review
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.
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.
eSolutions Manual - Powered by Cognero
40. Q(–2, 4), R(5, 6), S(12, 4), T(5, 2)
SOLUTION: Page 13
Since the products of the slopes of consecutive sides are -1, the sides are perpendicular.
So,Guide
the quadrilateral
is also a rectangle and a square.
Study
and Review
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.
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.
eSolutions Manual - Powered by Cognero
Page 14
, 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.
Study Guide and Review
Check whether the two diagonals are perpendicular.
Undefined slope and 0 slope are perpendicular, so the diagonals are perpendicular. It is a rhombus.
Find each measure.
41. GH
SOLUTION: Use the Pythagorean Theorem.
Since the length must be positive, GH = 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,
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?
eSolutions Manual - Powered by Cognero
Page 15
Study Guide and Review
So,
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.
eSolutions Manual - Powered by Cognero
Page 16