NAME ______________________________________________ DATE
8-1
____________ PERIOD _____
Skills Practice Angles of Polygons
Find the sum of the measures of the interior angles of each convex polygon. 1. nonagon
2. heptagon
3. decagon
4. 108
5. 120
Lesson 8-1
The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon. 6. 150
Find the measure of each interior angle using the given information. 7.
A
x
(2x 15)
8. L (2
B
(2x 15)
x
D
P
9. quadrilateral STUW with S T, U W, mS 2x 16, mU x 14
N
10. hexagon DEFGHI with D E G H, F I, mD 7x, mF 4x
T
D
E
I W
M
(2x 10)
2x
C
S
x 20) (3x 10)
U
F H
G
Find the measures of an interior angle and an exterior angle for each regular polygon. 11. quadrilateral
12. pentagon
13. dodecagon
Find the measures of an interior angle and an exterior angle given the number of sides of each regular polygon. Round to the nearest tenth if necessary. 14. 8
©
Glencoe/McGraw-Hill
15. 9
16. 13
419
Glencoe Geometry
NAME ______________________________________________ DATE
8-2
____________ PERIOD _____
Practice Parallelograms
Complete each statement about LMNP. Justify your answer. 1. L Q
M
Q
?
P
2. LMN
?
3. LMP
?
4. NPL is supplementary to 5. L M
L
N
? .
?
ALGEBRA Use RSTU to find each measure or value. 6. mRST
7. mSTU
8. mTUR
9. b
R
S
25
B
30 4b 1
23
U
T
COORDINATE GEOMETRY Find the coordinates of the intersection of the diagonals of parallelogram PRYZ given each set of vertices. 10. P(2, 5), R(3, 3), Y(2, 3), Z(3, 1)
11. P(2, 3), R(1, 2), Y(5, 7), Z(4, 2)
12. PROOF Write a paragraph proof of the following. Given: PRST and PQVU Prove: V S
Q
P U
V
T
13. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the paving stone for a sidewalk. If m1 is 130, find m2, m3, and m4.
©
Glencoe/McGraw-Hill
426
R S
1 4
2 3
Glencoe Geometry
NAME ______________________________________________ DATE
8-3
____________ PERIOD _____
Practice Tests for Parallelograms
Determine whether each quadrilateral is a parallelogram. Justify your answer. 1.
2.
3.
118
4.
62
62
118
COORDINATE GEOMETRY Determine whether a figure with the given vertices is a parallelogram. Use the method indicated. 5. P(5, 1), S(2, 2), F(1, 3), T(2, 2); Slope Formula
6. R(2, 5), O(1, 3), M(3, 4), Y(6, 2); Distance and Slope Formula
ALGEBRA Find x and y so that each quadrilateral is a parallelogram. 7.
(5x 29) (3y 15)
9.
(5y 9)
4 x
10.
(7x 11)
6x 7y 3
8.
12y 7
8 2y 2 3 x 5 4 3y
2x
y
4x 6
6
23
2 4y x 12
11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms?
©
Glencoe/McGraw-Hill
432
Glencoe Geometry
NAME ______________________________________________ DATE
8-4
____________ PERIOD _____
Practice Rectangles
ALGEBRA RSTU is a rectangle.
R
1. If UZ x 21 and ZS 3x 15, find US.
S
Z
U
T
2. If RZ 3x 8 and ZS 6x 28, find UZ. 3. If RT 5x 8 and RZ 4x 1, find ZT. 4. If mSUT 3x 6 and mRUS 5x 4, find mSUT. 5. If mSRT x2 9 and mUTR 2x 44, find x. 6. If mRSU x2 1 and mTUS 3x 9, find mRSU. GHJK is a rectangle. Find each measure if m1 37. 7. m2
G
8. m3 K
9. m4
10. m5
11. m6
12. m7
2
1 3
5 6
7 4
H
J
COORDINATE GEOMETRY Determine whether BGHL is a rectangle given each set of vertices. Justify your answer. 13. B(4, 3), G(2, 4), H(1, 2), L(1, 3)
14. B(4, 5), G(6, 0), H(3, 6), L(7, 1)
15. B(0, 5), G(4, 7), H(5, 4), L(1, 2)
16. LANDSCAPING Huntington Park officials approved a rectangular plot of land for a Japanese Zen garden. Is it sufficient to know that opposite sides of the garden plot are congruent and parallel to determine that the garden plot is rectangular? Explain.
©
Glencoe/McGraw-Hill
438
Glencoe Geometry
NAME ______________________________________________ DATE
8-5
____________ PERIOD _____
Practice Rhombi and Squares
Use rhombus PRYZ with RK 4y 1, ZK 7y 14, PK 3x 1, and YK 2x 6. 1. Find PY.
2. Find RZ.
3. Find RY.
4. Find mYKZ.
Use rhombus MNPQ with PQ 32 , PA 4x 1, and AM 9x 6. 5. Find AQ.
6. Find mAPQ.
7. Find mMNP.
8. Find PM.
Y
R
K Z
P
N
P A
M
Q
COORDINATE GEOMETRY Given each set of vertices, determine whether BEFG is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning. 9. B(9, 1), E(2, 3), F(12, 2), G(1, 4)
10. B(1, 3), E(7, 3), F(1, 9), G(5, 3)
11. B(4, 5), E(1, 5), F(7, 1), G(2, 1)
12. TESSELATIONS The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.
©
Glencoe/McGraw-Hill
444
Glencoe Geometry
NAME ______________________________________________ DATE
8-6
____________ PERIOD _____
Skills Practice
COORDINATE GEOMETRY ABCD is a quadrilateral with vertices A(4, 3), B(3, 3), C(6, 4), D(7, 4).
1. Verify that ABCD is a trapezoid.
2. Determine whether ABCD is an isosceles trapezoid. Explain.
COORDINATE GEOMETRY EFGH is a quadrilateral with vertices E(1, 3), F(5, 0), G(8, 5), H(4, 4). 3. Verify that EFGH is a trapezoid.
4. Determine whether EFGH is an isosceles trapezoid. Explain.
COORDINATE GEOMETRY LMNP is a quadrilateral with vertices L(1, 3), M(4, 1), N(6, 3), P(0, 7). 5. Verify that LMNP is a trapezoid. 6. Determine whether LMNP is an isosceles trapezoid. Explain.
ALGEBRA Find the missing measure(s) for the given trapezoid. 7. For trapezoid HJKL, S and T are midpoints of the legs. Find HJ. H
J 72
S L
W
K
86
42 35
Z
Q Y
19
10. For isosceles trapezoid QRST, find the length of the median, mQ, and mS. 60
T
E
S
U
T
85 G 14 F
©
X 12
P
T
9. For trapezoid DEFG, T and U are midpoints of the legs. Find TU, mE, and mG. D
8. For trapezoid WXYZ, P and Q are midpoints of the legs. Find WX.
Glencoe/McGraw-Hill
Q
449
125 25 R
Glencoe Geometry
Lesson 8-6
Trapezoids
NAME ______________________________________________ DATE
8-7
____________ PERIOD _____
Practice Coordinate Proof with Quadrilaterals
Position and label each quadrilateral on the coordinate plane. 1. parallelogram with side length b units and height a units
2. isosceles trapezoid with height b units, bases 2c a units and 2c a units
D ( c, a)
C ( b c, a )
D(a, b) C(2c, b)
A(0, 0)
B(b, 0)
A(0, 0)
B(a 2c, 0)
Name the missing coordinates for each quadrilateral. 3. parallelogram
4. isosceles trapezoid
y
y
L(c, ?)
K(?, a)
O H(0, 0)
J (2b, 0) x
Y(?, ?)
Z ( b, c)
X(?, ?) O
W(a, 0) x
Position and label the figure on the coordinate plane. Then write a coordinate proof for the following. 5. The opposite sides of a parallelogram are congruent.
6. THEATER A stage is in the shape of a trapezoid. Write a coordinate proof to prove that T R and S F are parallel.
D ( b, c)
C ( a b, c )
A(0, 0)
B(a, 0)
y
F(0, 25)
S(30, 25)
O T(10, 0) R(20, 0)
©
Glencoe/McGraw-Hill
456
x
Glencoe Geometry
©
Angles of Polygons
Skills Practice
____________ PERIOD _____
Glencoe/McGraw-Hill 900
2. heptagon
1440
3. decagon
6
5
12
6. 150
D
C
(2x ⫺ 15)⬚
x⬚
B
m⬔A ⫽ 115, m⬔B ⫽ 65, m⬔C ⫽ 115, m⬔D ⫽ 65
x⬚
(2x ⫺ 15)⬚
A
A3
T
U
P
(2x ⫺ 10)⬚
N
M
H
G
E F
m⬔D ⫽ 140, m⬔E ⫽ 140, m⬔F ⫽ 80, m⬔G ⫽ 140, m⬔H ⫽ 140, m⬔I ⫽ 80
I
D
10. hexagon DEFGHI with ⬔D ⬔E ⬔G ⬔H, ⬔F ⬔I, m⬔D ⫽ 7x, m⬔F ⫽ 4x
m⬔L ⫽ 100, m⬔M ⫽ 110, m⬔N ⫽ 70, m⬔P ⫽ 80
2x ⬚
x ⫹ 20)⬚ (3x ⫺ 10)⬚
8. L (2
108, 72
12. pentagon
150, 30
13. dodecagon
©
Glencoe/McGraw-Hill
135, 45
14. 8
140, 40
15. 9
419
Glencoe Geometry
Answers
Glencoe Geometry
152.3, 27.7
16. 13
Find the measures of an interior angle and an exterior angle given the number of sides of each regular polygon. Round to the nearest tenth if necessary.
90, 90
11. quadrilateral
Find the measures of an interior angle and an exterior angle for each regular polygon.
m⬔S ⫽ 116, m⬔T ⫽ 116, m⬔U ⫽ 64, m⬔W ⫽ 64
W
S
9. quadrilateral STUW with ⬔S ⬔T, ⬔U ⬔W, m⬔S ⫽ 2x ⫹ 16, m⬔U ⫽ x ⫹ 14
7.
Find the measure of each interior angle using the given information.
5. 120
4. 108
The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon.
1260
1. nonagon
Find the sum of the measures of the interior angles of each convex polygon.
8-1
NAME ______________________________________________ DATE
(Average)
Angles of Polygons
Practice
____________ PERIOD _____
2160
1620
2700
3. 17-gon
15
10
18
6. 160
(x ⫹ 15)⬚
(2x ⫹ 15)⬚
(3x ⫺ 20)⬚
K
x⬚
M
m⬔J ⫽ 115, m⬔K ⫽ 130, m⬔M ⫽ 50, m⬔N ⫽ 65
N
J
U
T
S
m⬔R ⫽ 128, m⬔S ⫽ 52, m⬔T ⫽ 128, m⬔U ⫽ 52
R
8. quadrilateral RSTU with m⬔R ⫽ 6x ⫺ 4, m⬔S ⫽ 2x ⫹ 8
165, 15
10. 24-gon
168, 12
11. 30-gon
163.6, 16.4
13. 22
171, 9
14. 40
©
Glencoe/McGraw-Hill
360 420
Glencoe Geometry
15. CRYSTALLOGRAPHY Crystals are classified according to seven crystal systems. The basis of the classification is the shapes of the faces of the crystal. Turquoise belongs to the triclinic system. Each of the six faces of turquoise is in the shape of a parallelogram. Find the sum of the measures of the interior angles of one such face.
154.3, 25.7
12. 14
Find the measures of an interior angle and an exterior angle given the number of sides of each regular polygon. Round to the nearest tenth if necessary.
157.5, 22.5
9. 16-gon
Find the measures of an interior angle and an exterior angle for each regular polygon. Round to the nearest tenth if necessary.
7.
Find the measure of each interior angle using the given information.
5. 156
4. 144
The measure of an interior angle of a regular polygon is given. Find the number of sides in each polygon.
2. 14-gon
1. 11-gon
Find the sum of the measures of the interior angles of each convex polygon.
8-1
NAME ______________________________________________ DATE
Answers (Lesson 8-1)
Lesson 8-1
©
Parallelograms
Skills Practice
Glencoe/McGraw-Hill
2. D E
3. G H
?
?
A6
60 10. a ⫽
15
8. m⬔WZY ⫽ Z
W
30
(1, 1)
Y
70⬚
12. H(⫺1, 4), J(3, 3), K(3, ⫺2), L(⫺1, ⫺1)
50⬚
A
2a
F
©
X
E
Glencoe/McGraw-Hill
425
Glencoe Geometry
Given: ⵥABCD A B Prove: ⬔A and ⬔B are supplementary. D C ⬔B and ⬔C are supplementary. ⬔C and ⬔D are supplementary. ⬔D and ⬔A are supplementary. Proof: We are given ⵥABCD, so we know that A 苶B 苶 || C 苶D 苶 and B 苶C 苶 || D 苶A 苶 by the definition of a parallelogram. We also know that if two parallel lines are cut by a transversal, then consecutive interior angles are supplementary. So, ⬔A and ⬔B, ⬔B and ⬔C, ⬔C and ⬔D, and ⬔D and ⬔A are pairs of supplementary angles.
13. PROOF Write a paragraph proof of the theorem Consecutive angles in a parallelogram are supplementary.
(3.5, 2)
11. H(1, 1), J(2, 3), K(6, 3), L(5, 1)
diagonals of parallelogram HJKL given each set of vertices.
COORDINATE GEOMETRY Find the coordinates of the intersection of the
9. m⬔WXY ⫽
7. m⬔XYZ ⫽ 120
60
䉭FEG ; diag. of ⵥ separates ⵥ into 2 ⬵ 䉭s.
ALGEBRA Use ⵥWXYZ to find each measure or value.
6. 䉭DGE
G
H
? . ⬔DEF or ⬔FGD ; cons. ⭄ in ⵥ are suppl.
⬔FGD ; opp. ⭄ of ⵥ are ⬵.
苶H E 苶 ; diag. of ⵥ bisect each other.
苶F G 苶; opp. sides of ⵥ are ⬵.
苶F E 苶; opp. sides of ⵥ are ||.
5. ⬔EFG is supplementary to
4. ⬔DEF
?
?
?
1. D G ||
D
____________ PERIOD _____
Complete each statement about ⵥDEFG. Justify your answer.
8-2
NAME ______________________________________________ DATE
(Average)
Parallelograms
Practice
?
3. 䉭LMP
?
苶P N 苶 ; opp. sides of ⵥ are ⬵.
9. b ⫽
6
7. m⬔STU ⫽
55
25⬚
U
30⬚ 4b ⫺ 1
R
(⫺1.5, ⫺2)
©
Glencoe/McGraw-Hill
426
13. CONSTRUCTION Mr. Rodriquez used the parallelogram at the right to design a herringbone pattern for a paving stone. He will use the paving stone for a sidewalk. If m⬔1 is 130, find m⬔2, m⬔3, and m⬔4.
Proof: We are given ⵥPRST and ⵥPQVU. Since opposite angles of a parallelogram are congruent, ⬔P ⬵ ⬔V and ⬔P ⬵ ⬔S. Since congruence of angles is transitive, ⬔V ⬵ ⬔S by the Transitive Property of Congruence.
50, 130, 50
B
S
4
P
1 3
Q V S
2
R
T
N
Glencoe Geometry
T
U
23
M
11. P(2, 3), R(1, ⫺2), Y(⫺5, ⫺7), Z(⫺4, ⫺2)
12. PROOF Write a paragraph proof of the following. Given: ⵥPRST and ⵥPQVU Prove: ⬔V ⬔S
(0, 1)
10. P(2, 5), R(3, 3), Y(⫺2, ⫺3), Z(⫺3, ⫺1)
diagonals of parallelogram PRYZ given each set of vertices.
COORDINATE GEOMETRY Find the coordinates of the intersection of the
8. m⬔TUR ⫽ 125
6. m⬔RST ⫽ 125
ALGEBRA Use ⵥRSTU to find each measure or value.
5. L M
P
Q
? . ⬔MNP or ⬔PLM ; cons. ⭄ in ⵥ are suppl.
䉭NPM; diag. of ⵥ separates ⵥ into 2 ⬵ 䉭s.
⬔NPL; opp. ⭄ of ⵥ are ⬵.
4. ⬔NPL is supplementary to
?
苶Q N 苶 ; diag. of ⵥ bisect each other.
2. ⬔LMN
1. LQ ?
L
____________ PERIOD _____
Complete each statement about ⵥLMNP. Justify your answer.
8-2
NAME ______________________________________________ DATE
Answers (Lesson 8-2)
Glencoe Geometry
Lesson 8-2
©
Tests for Parallelograms
Skills Practice
____________ PERIOD _____
Glencoe/McGraw-Hill
No; none of the tests for parallelograms is fulfilled.
Yes; a pair of opposite sides is parallel and congruent.
4.
2.
Yes; both pairs of opposite sides are congruent.
Yes; both pairs of opposite angles are congruent.
A9
©
y ⫹ 19
(3x ⫹ 10)⬚
Glencoe/McGraw-Hill
x ⫽ 45, y ⫽ 20
(2y ⫺ 5)⬚
(4x ⫺ 35)⬚
(y ⫹ 15)⬚
431
11.
9. 3x 3
3x ⫺ 14
x ⫹ 20
Glencoe Geometry
Glencoe Geometry
Answers
y ⫹ 20
x ⫽ 17, y ⫽ 9
3y ⫹ 2
x ⫽ 3, y ⫽ 14
y⫹
11 2y ⫺
3
x ⫽ 24, y ⫽ 19
x ⫹ 16
2y
2x ⫺ 8
⫺ 4x
10.
8.
ALGEBRA Find x and y so that each quadrilateral is a parallelogram.
no
7. W(2, 5), R(3, 3), Y(⫺2, ⫺3), N(⫺3, 1); Midpoint Formula
yes
6. S(⫺2, 1), R(1, 3), T(2, 0), Z(⫺1, ⫺2); Distance and Slope Formula
yes
5. P(0, 0), Q(3, 4), S(7, 4), Y(4, 0); Slope Formula
parallelogram. Use the method indicated.
COORDINATE GEOMETRY Determine whether a figure with the given vertices is a
3.
1.
Determine whether each quadrilateral is a parallelogram. Justify your answer.
8-3
NAME ______________________________________________ DATE
(Average)
Tests for Parallelograms
Practice
____________ PERIOD _____
118⬚ 118⬚
62⬚
Yes; both pairs of opposite angles are congruent.
62⬚
Yes; the diagonals bisect each other. 4.
2.
No; none of the tests for parallelograms is fulfilled.
No; none of the tests for parallelograms is fulfilled.
(7x ⫺ 11)⬚
(5y ⫺ 9)⬚
12y ⫺ 7 ⫺4x ⫹ 6
⫺6x
x ⫽ ⫺3, y ⫽ 2
7y ⫹ 3
x ⫽ 20, y ⫽ 12
(3y ⫹ 15)⬚
(5x ⫹ 29)⬚
10.
8.
2x
6
⫺2 ⫺4y x⫹ 12
x ⫽ ⫺2, y ⫽ ⫺5
23
⫹
y⫹
⫺
x ⫽ ⫺6, y ⫽ 13
2
8 2y ⫹ ⫺3 x⫹ 5 4 3y ⫺
⫺4 x⫺
©
Glencoe/McGraw-Hill
432
Glencoe Geometry
Sample answer: Confirm that both pairs of opposite ⬔s are ⬵.
11. TILE DESIGN The pattern shown in the figure is to consist of congruent parallelograms. How can the designer be certain that the shapes are parallelograms?
9.
7.
ALGEBRA Find x and y so that each quadrilateral is a parallelogram.
yes
6. R(⫺2, 5), O(1, 3), M(⫺3, ⫺4), Y(⫺6, ⫺2); Distance and Slope Formula
no
5. P(⫺5, 1), S(⫺2, 2), F(⫺1, ⫺3), T(2, ⫺2); Slope Formula
COORDINATE GEOMETRY Determine whether a figure with the given vertices is a parallelogram. Use the method indicated.
3.
1.
Determine whether each quadrilateral is a parallelogram. Justify your answer.
8-3
NAME ______________________________________________ DATE
Answers (Lesson 8-3)
Lesson 8-3
©
Rectangles
Skills Practice
Glencoe/McGraw-Hill
A12 12. m⬔5 40 14. m⬔7 50 16. m⬔9 80
11. m⬔4 50
13. m⬔6 40
15. m⬔8 100
T
P 2 9 8 6
7 5
3 4
R
S
©
Glencoe/McGraw-Hill
437
Glencoe Geometry
Yes; sample answer: Opposite sides are parallel and consecutive sides are perpendicular.
19. T(4, 1), U(3, ⫺1), X(⫺3, 2), Y(⫺2, 4)
Yes; sample answer: Opposite sides are congruent and diagonals are congruent.
18. T(⫺6, 3), U(0, 6), X(2, 2), Y(⫺4, ⫺1)
No; sample answer: Angles are not right angles.
17. T(⫺3, ⫺2), U(⫺4, 2), X(2, 4), Y(3, 0)
of vertices. Justify your answer.
COORDINATE GEOMETRY Determine whether TUXY is a rectangle given each set
10. m⬔3 40
9. m⬔2 40
PRST is a rectangle. Find each measure if m⬔1 ⫽ 50.
8. If m⬔BAC ⫽ x2 ⫹ 3 and m⬔CAD ⫽ x ⫹ 15, find m⬔BAC. 67 or 84
12. m⬔7 74
11. m⬔6 106
K
2
3
1
Z
6
7 4
T
S
5
J
H
©
Glencoe/McGraw-Hill
438
Glencoe Geometry
No; if you only know that opposite sides are congruent and parallel, the most you can conclude is that the plot is a parallelogram.
16. LANDSCAPING Huntington Park officials approved a rectangular plot of land for a Japanese Zen garden. Is it sufficient to know that opposite sides of the garden plot are congruent and parallel to determine that the garden plot is rectangular? Explain.
No; sample answer: Diagonals are not congruent.
15. B(0, 5), G(4, 7), H(5, 4), L(1, 2)
Yes; sample answer: Opposite sides are congruent and diagonals are congruent.
14. B(⫺4, 5), G(6, 0), H(3, ⫺6), L(⫺7, ⫺1)
Yes; sample answer: Opposite sides are parallel and consecutive sides are perpendicular.
13. B(⫺4, 3), G(⫺2, 4), H(1, ⫺2), L(⫺1, ⫺3)
COORDINATE GEOMETRY Determine whether BGHL is a rectangle given each set of vertices. Justify your answer.
10. m⬔5 53
8. m⬔3 37
9. m⬔4 37
7. m⬔2 53
G
6. If m⬔RSU ⫽ x2 ⫺ 1 and m⬔TUS ⫽ 3x ⫹ 9, find m⬔RSU. 24 or 3
6. If m⬔BDC ⫽ 7x ⫹ 1 and m⬔ADB ⫽ 9x ⫺ 7, find m⬔BDC. 43
GHJK is a rectangle. Find each measure if m⬔1 ⫽ 37.
5. If m⬔SRT ⫽ x2 ⫹ 9 and m⬔UTR ⫽ 2x ⫹ 44, find x. ⫺5 or 7
5. If m⬔DAC ⫽ 2x ⫹ 4 and m⬔BAC ⫽ 3x ⫹ 1, find x. 17
7. If m⬔ABD ⫽ x2 ⫺ 7 and m⬔CDB ⫽ 4x ⫹ 5, find x. 6
4. If m⬔SUT ⫽ 3x ⫹ 6 and m⬔RUS ⫽ 5x ⫺ 4, find m⬔SUT. 39
U
4. If DE ⫽ 6x ⫺ 7 and AE ⫽ 4x ⫹ 9, find DB. 82
1. If UZ ⫽ x ⫹ 21 and ZS ⫽ 3x ⫺ 15, find US. 78
R
3. If RT ⫽ 5x ⫹ 8 and RZ ⫽ 4x ⫹ 1, find ZT. 9
C
(Average)
ALGEBRA RSTU is a rectangle.
Rectangles
Practice
____________ PERIOD _____
3. If AE ⫽ 3x ⫹ 3 and EC ⫽ 5x ⫺ 15, find AC. 60
1
E
B
8-4
NAME ______________________________________________ DATE
2. If RZ ⫽ 3x ⫹ 8 and ZS ⫽ 6x ⫺ 28, find UZ. 44
D
A
____________ PERIOD _____
Lesson 8-4
2. If AC ⫽ x ⫹ 3 and DB ⫽ 3x ⫺ 19, find AC. 14
1. If AC ⫽ 2x ⫹ 13 and DB ⫽ 4x ⫺ 1, find x. 7
ALGEBRA ABCD is a rectangle.
8-4
NAME ______________________________________________ DATE
Answers (Lesson 8-4)
Glencoe Geometry
©
Rhombi and Squares
Skills Practice
Glencoe/McGraw-Hill 13
4. Find DM.
5
2. Find AL.
A15 60
10. Find m⬔SRV.
60
8. Find m⬔SVT.
12
6. Find TV. R
M
D
S
A
N
L
V
K
T
____________ PERIOD _____
©
Glencoe/McGraw-Hill
443
Glencoe Geometry
Answers
Glencoe Geometry
None; opposite sides are congruent, but the diagonals are neither congruent nor perpendicular.
14. Q(2, ⫺4), R(⫺6, ⫺8), S(⫺10, 2), T(⫺2, 6)
Rhombus; all sides are congruent and the diagonals are perpendicular, but not congruent.
13. Q(⫺6, ⫺1), R(4, ⫺6), S(2, 5), T(⫺8, 10)
Rhombus; all sides are congruent and the diagonals are perpendicular, but not congruent.
12. Q(⫺5, 12), R(5, 12), S(⫺1, 4), T(⫺11, 4)
Rhombus, rectangle, square; all sides are congruent and the diagonals are perpendicular and congruent.
11. Q(3, 5), R(3, 1), S(⫺1, 1), T(⫺1, 5)
is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.
COORDINATE GEOMETRY Given each set of vertices, determine whether ⵥQRST
120
9. Find m⬔RST.
30
7. Find m⬔NTV.
2
5. Find y.
Use rhombus RSTV with RS ⫽ 5y ⫹ 2, ST ⫽ 3y ⫹ 6, and NV ⫽ 6.
90
3. Find m⬔KAL.
3
1. Find x.
Use rhombus DKLM with AM ⫽ 4x, AK ⫽ 5x ⫺ 3, and DL ⫽ 10.
8-5
NAME ______________________________________________ DATE
(Average)
Rhombi and Squares
Practice
90
4. Find m⬔YKZ.
42
2. Find RZ.
6
8. Find PM.
45
6. Find m⬔APQ.
M
N
R
P
A
K Z
Q
P
Y
____________ PERIOD _____
©
Glencoe/McGraw-Hill
444
The figure consists of 6 congruent rhombi.
12. TESSELATIONS The figure is an example of a tessellation. Use a ruler or protractor to measure the shapes and then name the quadrilaterals used to form the figure.
None; two of the opposite sides are not congruent.
11. B(⫺4, ⫺5), E(1, ⫺5), F(⫺7, ⫺1), G(⫺2, ⫺1)
Glencoe Geometry
Rhombus, rectangle, square; all sides are congruent and the diagonals are perpendicular and congruent.
10. B(1, 3), E(7, ⫺3), F(1, ⫺9), G(⫺5, ⫺3)
Rhombus; all sides are congruent and the diagonals are perpendicular, but not congruent.
9. B(⫺9, 1), E(2, 3), F(12, ⫺2), G(1, ⫺4)
is a rhombus, a rectangle, or a square. List all that apply. Explain your reasoning.
COORDINATE GEOMETRY Given each set of vertices, determine whether ⵥBEFG
90
7. Find m⬔MNP.
3
5. Find AQ.
Use rhombus MNPQ with PQ ⫽ 3兹2 苶, PA ⫽ 4x ⫺ 1, and AM ⫽ 9x ⫺ 6.
29
3. Find RY.
40
1. Find PY.
Use rhombus PRYZ with RK ⫽ 4y ⫹ 1, ZK ⫽ 7y ⫺ 14, PK ⫽ 3x ⫺ 1, and YK ⫽ 2x ⫹ 6.
8-5
NAME ______________________________________________ DATE
Answers (Lesson 8-5)
Lesson 8-5
©
Trapezoids
Skills Practice
____________ PERIOD _____
Glencoe/McGraw-Hill (Average)
A18 || N 苶P 苶
©
86
72
J
T
K
T
35⬚
85⬚
U
E
G 14 F
42
Glencoe/McGraw-Hill
28, 95, 145
D
9. For trapezoid DEFG, T and U are midpoints of the legs. Find TU, m⬔E, and m⬔G.
L
S
H
7. For trapezoid HJKL, S and T are midpoints of the legs. Find HJ. 58
19
12
X Q Y
449
Q
125⬚ 25 R
60
42.5, 125, 55
T
S
Glencoe Geometry
10. For isosceles trapezoid QRST, find the length of the median, m⬔Q, and m⬔S.
Z
P
W
8. For trapezoid WXYZ, P and Q are midpoints of the legs. Find WX. 5
ALGEBRA Find the missing measure(s) for the given trapezoid.
not isosceles; LP ⫽ 兹17 苶 and MN ⫽ 兹8 苶
6. Determine whether LMNP is an isosceles trapezoid. Explain.
5. Verify that LMNP is a trapezoid. 苶 LM 苶
N(⫺6, 3), P(0, 7).
COORDINATE GEOMETRY LMNP is a quadrilateral with vertices L(⫺1, 3), M(⫺4, 1),
not isosceles; EH ⫽ 兹26 苶 and FG ⫽ 兹34 苶
4. Determine whether EFGH is an isosceles trapezoid. Explain.
苶F E 苶 || G 苶H 苶
3. Verify that EFGH is a trapezoid.
F 28
18
E Y D
140⬚ 36
21
125⬚
H M I
66 42
L
C
R
T
Y 5Z
34 60⬚
V
19.5, 60, 120
8. For isosceles trapezoid TVZY, find the length of the median, m⬔T, and m⬔Z.
P
B
W
6. For trapezoid WRLP, B and C are midpoints of the legs. Find LP. 18
©
Glencoe/McGraw-Hill
450
Sample answer: the measures of the base angles Glencoe Geometry
10. DESK TOPS A carpenter needs to replace several trapezoid-shaped desktops in a classroom. The carpenter knows the lengths of both bases of the desktop. What other measurements, if any, does the carpenter need?
9. CONSTRUCTION A set of stairs leading to the entrance of a building is designed in the shape of an isosceles trapezoid with the longer base at the bottom of the stairs and the shorter base at the top. If the bottom of the stairs is 21 feet wide and the top is 14, find the width of the stairs halfway to the top. 17.5 ft
F
K
G
7. For trapezoid FGHI, K and M are midpoints of the legs. Find FI, m⬔F, and m⬔I. 51, 40, 55
C
V
5. For trapezoid CDEF, V and Y are midpoints of the legs. Find CD. 38
ALGEBRA Find the missing measure(s) for the given trapezoid.
not isosceles; BJ ⫽ 兹50 苶 and GH ⫽ 兹125 苶
4. Determine whether BGHJ is an isosceles trapezoid. Explain.
苶G B 苶 || H 苶J 苶
3. Verify that BGHJ is a trapezoid.
COORDINATE GEOMETRY BGHJ is a quadrilateral with vertices B(⫺9, 1), G(2, 3), H(12, ⫺2), J(⫺10, ⫺6).
not isosceles; RU ⫽ 兹37 苶 and ST ⫽ 兹34 苶
2. Determine whether RSTU is an isosceles trapezoid. Explain.
苶S R 苶 || T 苶U 苶
1. Verify that RSTU is a trapezoid.
T(10, ⫺2), U(⫺4, ⫺9).
Trapezoids
Practice
____________ PERIOD _____
COORDINATE GEOMETRY RSTU is a quadrilateral with vertices R(⫺3, ⫺3), S(5, 1),
8-6
NAME ______________________________________________ DATE
G(8, ⫺5), H(⫺4, 4).
Lesson 8-6
COORDINATE GEOMETRY EFGH is a quadrilateral with vertices E(1, 3), F(5, 0),
isosceles; AD ⫽ 兹58 苶 and BC ⫽ 兹58 苶
2. Determine whether ABCD is an isosceles trapezoid. Explain.
苶B A 苶 || C 苶D 苶
1. Verify that ABCD is a trapezoid.
B(3, ⫺3), C(6, 4), D(⫺7, 4).
COORDINATE GEOMETRY ABCD is a quadrilateral with vertices A(⫺4, ⫺3),
8-6
NAME ______________________________________________ DATE
Answers (Lesson 8-6)
Glencoe Geometry
©
Coordinate Proof with Quadrilaterals
Skills Practice
____________ PERIOD _____
Glencoe/McGraw-Hill
B(2a, 0)
O A(0, 0)
A21
x
O T(0, 0)
O S(a, 0)
Y(?, ?)
y
T(b, 0)
W (?, c)
x
6. isosceles trapezoid
E(5b, 0) x
F(5b, 2b)
W (a ⫹ b, c), Y (0, c)
G(0, 2b)
B(b ⫹ c, 0) x
C(c, a)
©
Q
M O
y
undefined
Glencoe/McGraw-Hill
455
Glencoe Geometry
Answers
Glencoe Geometry
MN 苶 苶 and Q 苶P 苶 have the same slope. M 苶Q 苶 and N 苶P 苶 also have the same slope. 苶N M 苶 is perpendicular to M 苶Q 苶. Therefore, both pairs of opposite sides are parallel and consecutive sides are perpendicular. This means that MNPQ is a rectangle.
slope of M 苶N 苶 ⫽ ᎏᎏ or 0
x
C(2a, 0)
D(0, ⫺2b)
P
B(0, 2b) N
⫺b ⫺ (⫺b) b⫺b slope of 苶 QP 苶 ⫽ ᎏᎏ or 0 ⫺a ⫺ a a ⫺ (⫺a) b ⫺ (⫺b) ⫺b ⫺ b 苶苶 Q ⫽ ᎏᎏ or undefined slope of N 苶P 苶 ⫽ ᎏᎏ or slope of M a⫺a ⫺a ⫺ (⫺a)
Given: ABCD is a rhombus. M, N, P, and Q are the midpoints 苶B 苶, B 苶C 苶, C 苶D 苶, and D 苶A 苶, respectively. of A A(⫺2a, 0) Prove: MNPQ is a rectangle. Proof: M ⫽ (⫺a, b), N ⫽ (a, b), P ⫽ (a, ⫺b), and Q ⫽ (⫺a, ⫺b).
7. The segments joining the midpoints of the sides of a rhombus form a rectangle.
Position and label the figure on the coordinate plane. Then write a coordinate proof for the following.
S(a, ?)
R(?, ?)
R (a ⫹ b, c), S (a, 0)
P(b, c)
y
5. parallelogram
O D(0, 0)
U(c, 0)
O A(0, 0)
x
y
G(?, ?)
4. rectangle
C(?, ?)
D(0, a)
C (c, a)
y
3. rectangle
Name the missing coordinates for each quadrilateral.
O A(0, 0)
B(b, a)
x
y
C(2a, a)
D(0, a)
2. isosceles trapezoid with height a units, bases c ⫺ b units and b ⫹ c units
y
1. rectangle with length 2a units and height a units
Position and label each quadrilateral on the coordinate plane.
8-7
NAME ______________________________________________ DATE
(Average)
Coordinate Proof with Quadrilaterals
Practice
____________ PERIOD _____
B(b, 0) x
C(b ⫹ c, a)
O A(0, 0)
J (2b, 0) x
K(?, a)
y
W(a, 0) x
Z(b, c)
X (⫺a, 0), Y (⫺b, c)
X(?, ?) O
Y(?, ?)
4. isosceles trapezoid
©
⫽
兹苶 a2
or a
Glencoe/McGraw-Hill
456
苶F S 苶 both have a slope of 0, T 苶R 苶 || S 苶F 苶.
Proof: The slope of T 苶R 苶 ⫽ ᎏᎏ ⫽ ᎏᎏ and
0⫺0 0 20 ⫺ 10 10 25 ⫺ 25 0 the slope of SF ⫽ ᎏᎏ ⫽ ᎏᎏ. Since T 苶R 苶 and 30 ⫺ 0 30
Given: T (10, 0), R (20, 0), S (30, 25), F (0, 25) 苶R 苶 || S 苶F 苶 Prove: T
6. THEATER A stage is in the shape of a trapezoid. Write a coordinate proof to prove that T R and S F are parallel.
AB ⫽ 兹苶 [(a ⫹ 苶 b) ⫺ a苶 ]2⫹ (c苶 ⫺ 0)2 ⫽ 兹苶 b2 ⫹ 苶 c2 AB ⫽ CD and AD ⫽ BC, so A 苶B 苶⬵C 苶D 苶 and A 苶D 苶⬵B 苶C 苶
(b ⫺ 0苶 )2 ⫹ (苶 c ⫺ 0苶 )2 ⫽ 兹苶 b2 ⫹ 苶 c2 AD ⫽ 兹苶
CD ⫽
兹苶 [(a ⫹ 苶 b) ⫺ b苶 ]2 ⫹ (苶 c ⫺ c苶 )2
AB ⫽ 兹苶 (a ⫺ 0苶 )2 ⫹ (苶 0 ⫺ 0苶 )2 ⫽ 兹苶 a 2 or a
Given: ABCD is a parallelogram. 苶B 苶⬵C 苶D 苶, A 苶D 苶⬵B 苶C 苶 Prove: A Proof:
5. The opposite sides of a parallelogram are congruent.
Glencoe Geometry
x
S(30, 25)
B(a, 0) x
C(a ⫹ b, c)
O T(10, 0) R(20, 0)
F(0, 25)
y
O A(0, 0)
D(b, c)
y
Position and label the figure on the coordinate plane. Then write a coordinate proof for the following.
K (2b ⫹ c, a), L (c, a)
O H(0, 0)
L(c, ?)
y
3. parallelogram
B(a ⫹ 2c, 0) x
D(a, b) C(2c, b)
y
2. isosceles trapezoid with height b units, bases 2c ⫺ a units and 2c ⫹ a units
Name the missing coordinates for each quadrilateral.
O A(0, 0)
D(c, a)
y
1. parallelogram with side length b units and height a units
Position and label each quadrilateral on the coordinate plane.
8-7
NAME ______________________________________________ DATE
Answers (Lesson 8-7)
Lesson 8-7