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Study Guide and Review
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
.
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
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15. ≈ 166.15
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Substitute n = 6 in
.
Study Guide and Review
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.
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.
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19. SOLUTION: Page 2
17. AD
SOLUTION: WeGuide
knowand
that Review
opposite sides of a parallelogram are congruent.
Study
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
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
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Determine whether each quadrilateral is a parallelogram. Justify your answer.
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5y = 60
y = 12
Study
So,Guide
x = 5and
andReview
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
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.
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by Cognero
YouManual
need to
walk through
the
Page 4
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.
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
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72 = 13x
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52 = 13x
4 = x
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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
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.
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.
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FH =
FK +- Powered
KH Diagonals
of a rectangle bisect each other.
= FK + FK FK = KH, substitution
= 32 + 32 Substitute.
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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.
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Since the length must be positive, AE = 7.9.
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= 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, 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)
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SOLUTION: First graph the quadrilateral.
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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, 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.
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Find each measure.
41. GH
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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
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,
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