Download Find the sum of the measures of the interior angles of each convex

Substitute n = 15 in
.
Study Guide and Review
Find the sum of the measures of the interior angles of each
convex polygon.
11. decagon
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 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
= 6 in by Cognero .
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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
Use
ABCD to find each measure.
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19. Study Guide and Review
Use
ABCD to find each measure.
SOLUTION: We know that opposite angles of a parallelogram are congruent.
So, ALGEBRA Find the value of each variable in each parallelogram.
20. 16. SOLUTION: We know that consecutive angles in a parallelogram are supplementary.
So,
Substitute.
SOLUTION: Since the opposite sides of a parallelogram are congruent, 2x + 9 = 4x –
5.
Solve for x.
2x + 9 = 4x – 5 Opp. sides of a parallelogram are .
9 = 2x – 5 Subtract 2x from each side.
14 = 2x Add 5 to each side.
7 = x Divide each side by 2.
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. Since the sides of a parallelogram are parallel, the alternate interior
angles are congruent. Thus, the alternate interior angles at top and
bottom must both have a measure of 4y as shown . Since the opposite
sides of the parallelogram are parallel, the consecutive interior angles
must be supplementary. So, set the sum of 42, 23, 4y, and 83 equal to
180 and solve for y.
SOLUTION: We know that opposite angles of a parallelogram are congruent.
So, ALGEBRA Find the value of each variable in each parallelogram.
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SOLUTION: So, x = 7 and y = 8.
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Since the opposite angles are congruent, 2x + 41 = 115.
Solve for x.
2x + 41 = 115
2x = 74
x = 37
Study
So,Guide
x = 7and
andReview
y = 8.
ALGEBRA Find x and y so that the quadrilateral is a
parallelogram.
21. SOLUTION: 26. Since the opposite sides are congruent, 3y + 13 = 2y + 19.
Solve for y.
3y + 13 = 2y + 19
y =6
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
Since the opposite angles are congruent, 2x + 41 = 115.
Solve for x.
2x + 41 = 115
2x = 74
x = 37
Solve for y.
3y + 36 = 9y - 12
36 = 6y - 12
48 = 6y
8 = y
ALGEBRA Find x and y so that the quadrilateral is a
parallelogram.
When x = 4 and y = 8 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
27. SOLUTION: We know that diagonals of a parallelogram bisect each other.
So,
.
Solve for x.
Solve for y.
3y +Manual
36 = 9y
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36 = 6y - 12
48 = 6y
Alternate interior angles in a parallelogram are congruent.
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48 = 6y
8 = y
Study
Guide
Review
When
x =and
4 and
y = 8 the quadrilateral is a parallelogram.
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)?
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)?
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 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
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Substitute
= 8 in 5x
20.
5x + 20 = 5(8) + 20
, find
.
SOLUTION: All four angles of a rectangle are right angles. So,
Substitute.
30. If SOLUTION: , find
.
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5x + 20 = 5(8) + 20
= 60
Study
Guide and Review
So, the length of the space is 60 inches.
ALGEBRA Quadrilateral EFGH is a rectangle.
= 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,
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.
= 64 Add.
EG is the same length as FH so EG = 64 feet.
32. Find
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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
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Theorem.
= 4(3) – 6 x = 3
= 12 – 6 Multiply.
= 6 Subtract.
Guide and Review
Study
35. SOLUTION: All the four sides of a rhombus are congruent. So,
triangle. Therefore,
So, EF = 6.
ALGEBRA ABCD is a rhombus. If EB = 9, AB = 12 and
, find each measure.
is an isosceles 36. CE
SOLUTION: The diagonals of a rhombus are perpendicular. Use AE to find CE.
Use the Pythagorean Theorem.
34. AE
SOLUTION: The diagonals of a rhombus are perpendicular. So, 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,
Since the length must be positive, AE = 7.9.
35. SOLUTION: All the four sides of a rhombus are congruent. So,
triangle. Therefore,
All the four sides of a rhombus are congruent. So,
triangle. Therefore,
is an isosceles 36. CE
is an isosceles 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: The diagonals of a rhombus are perpendicular. Use AE to find CE.
Use the Pythagorean Theorem.
SOLUTION: A rhombus is a parallelogram with all four sides congruent. So, FG = FJ
= 2.5 cm.
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Since the length must be positive, AE = 7.9.
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Find each measure.
41. GH
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
triangle. Therefore,
Since the length must be positive, GH = 19.2.
is an isosceles 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.
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
.
Find each measure.
41. GH
So,
SOLUTION: Use the Pythagorean Theorem.
Since the length must be positive, GH = 19.2.
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