Download Topic 6 Polygons and Quadrilaterals

Topic 6
Polygons and Quadrilaterals
TOPIC OVERVIEW
VOCABULARY
6-1
The Polygon Angle-Sum Theorems
English/Spanish Vocabulary Audio Online:
EnglishSpanish
equiangular polygon, p. 249
polígono equiángulo
equilateral polygon, p. 249
polígono equilátero
isosceles trapezoid, p. 281
trapecio isósceles
kite, p. 282cometa
midsegment of a trapezoid, p. 281segmento medio de un trapecio
parallelogram, p. 255paralelogramo
rectangle, p. 269rectángulo
regular polygon, p. 249
polígono regular
rhombus, p. 269rombo
square, p. 269cuadrado
trapezoid, p. 281trapecio
6-2
Properties of Parallelograms
6-3
Proving That a Quadrilateral
Is a Parallelogram
6-4
Properties of Rhombuses,
Rectangles, and Squares
6-5
Conditions for Rhombuses,
Rectangles, and Squares
6-6
Trapezoids and Kites
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246
Topic 6 Polygons and Quadrilaterals
3--Act Math
The Mystery
Sides
Have you every looked closely
at honeycombs? What shape
are they? How do you know?
Most often the cells in the
honeycombs look like hexagons,
but they might also look like
circles. Scientists now believe
that the bees make circular cells
that become hexagonal due to
the bees’ body heat and natural
physical forces.
What are some strategies you
use to identify shapes? Think
about this as you watch the
3-Act Math video.
Scan page to see a video
for this 3-Act Math Task.
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247
Technology Lab
Use With Lesson 6-1
Exterior Angles of Polygons
teks (5)(A), (1)(E)
Use geometry software. Construct a polygon similar to the one at the right.
Extend each side as shown. Mark a point on each ray so that you can
measure the exterior angles.
Use your figure to explore properties of a polygon.
• Measure each exterior angle.
• Calculate the sum of the measures of the exterior angles.
• Manipulate the polygon. Observe the sum of the measures of
the exterior angles of the new polygon.
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Exercises
1.Write a conjecture about the sum of the measures of the exterior angles (one at
each vertex) of a convex polygon. Test your conjecture with another polygon.
2.The figures below show a polygon that is decreasing in size until it finally
becomes a point. Describe how you could use this to justify your conjecture
in Exercise 1.
1
5
2
4
4
5
3
1
2
4 5
3 2 1
3
3.The figure at the right shows a square that has been copied several times.
Notice that you can use the square to completely cover, or tile, a plane, without
gaps
or overlaps.
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a.
Using geometry software, make several copies of other regular polygons with
3, 5, 6, and 8 sides. Regular polygons have sides of equal length and angles of
equal measure.
b.
Which of the polygons you made can tile a plane?
c.
Measure one exterior angle of each polygon (including the square).
d.
Write a conjecture about the relationship between the measure of an exterior
angle and your ability to tile a plane with the polygon. Test your conjecture
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with another regular polygon.
248
Technology Lab Exterior Angles of Polygons
6-1 The Polygon Angle-Sum Theorems
TEKS FOCUS
VOCABULARY
TEKS (5)(A) Investigate patterns to make conjectures about
geometric relationships, including angles formed by parallel lines
cut by a transversal, criteria required for triangle congruence, special
segments of triangles, diagonals of quadrilaterals, interior and
exterior angles of polygons, and special segments and angles of circles
choosing from a variety of tools.
•Equiangular polygon – An equiangular
TEKS (1)(C) Select tools, including real objects, manipulatives, paper and
pencil, and technology as appropriate, and techniques, including mental
math, estimation, and number sense as appropriate, to solve problems.
a polygon that is both equilateral and
equiangular.
polygon is a polygon with all angles
congruent.
•Equilateral polygon – An equilateral polygon
is a polygon with all sides congruent.
•Regular polygon – A regular polygon is
•Number sense – the understanding of what
Additional TEKS (1)(E), (1)(F)
numbers mean and how they are related
ESSENTIAL UNDERSTANDING
The sum of the interior angle measures of a polygon depends on the number of sides
the polygon has.
Key Concept Classifying Polygons Based on Sides and Angles
An equilateral polygon
is a polygon with all
sides congruent.
An equiangular polygon
is a polygon with all
angles congruent.
A regular polygon is
a polygon that is both
equilateral and equiangular.
Theorem 6-1 Polygon
Angle-Sum Theorem
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The sum of the measures of the interior angles of an n-gon is (n - 2)180.
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For a proof of Theorem 6-1, see the Reference section on page 683.
Corollary to the Polygon Angle-Sum Theorem
The measure of each interior angle of a regular n-gon is
(n - 2)180
.
n
You will prove the Corollary to the Polygon Angle-Sum Theorem in Exercise 16.
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249
Theorem 6-2 Polygon Exterior Angle-Sum Theorem
The sum of the measures of the exterior angles of a polygon, one at
each vertex, is 360.
For the pentagon, m∠1 + m∠2 + m∠3 + m∠4 + m∠5 = 360.
3
2
4
1
5
You will prove Theorem 6-2 in Exercise 9.
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TEKS
Process Standard (1)(C)
Problem 1
Investigating Interior Angles of Polygons
Choose from among a variety of tools (such as a ruler, a compass, or geometry
A software) to investigate the sums of the measures of the interior angles of
different polygons. Explain your choice.
Geometry software is a good way to identify the measures of the interior angles of
polygons. You can quickly make many different polygons and use the software to
find the measures of their angles.
B Use geometry software to make several triangles, quadrilaterals, pentagons,
and hexagons. Then complete the table.
How can recording
data in a table
help you make a
conjecture?
Recording data in a table
is an organized way to
present and analyze
information. You can look
for patterns in the data
and make a conjecture.
250
Sum of Interior Angle
Measures
Polygon
Triangle 1
Sum of Interior Angle
Measures
180
Polygon
Pentagon 1
Triangle 2
180
Pentagon 2
540
Triangle 3
180
Pentagon 3
540
Quadrilateral 1
360
Hexagon 1
720
Quadrilateral 2
360
Hexagon 2
720
Quadrilateral 3
360
Hexagon 3
720
540
C Use the data in the table in part B to make a conjecture about the sum of the
measures of the interior angles of a polygon.
Notice that the numbers in the table are all multiples of 180. Look at the patterns:
# 180 = 180Pentagon3# 180 = 540
Quadrilateral2 # 180 = 360
Hexagon 4 # 180 = 720
Triangle
1
Conjecture: If you subtract 2 from the number of sides and multiply by 180, you
will get the sum of the measures of the interior angles of any polygon.
Lesson 6-1 The Polygon Angle-Sum Theorems
Problem 2
Finding a Polygon Angle Sum
How many sides does
a heptagon have?
A heptagon has 7 sides.
What is the sum of the interior angle measures of a heptagon?
Sum = (n - 2)180
= (7 - 2)180
=5
= 900
# 180
Polygon Angle-Sum Theorem
Substitute 7 for n.
Simplify.
The sum of the interior angle measures of a heptagon is 900.
Problem 3
Using the Polygon Angle-Sum Theorem
How does the word
regular help you
answer the question?
The word regular tells
you that each angle has
the same measure.
STEM
Biology The common housefly, Musca domestica, has eyes
that consist of approximately 4000 facets. Each facet is
a regular hexagon. What is the measure of each
interior angle in one hexagonal facet?
(n - 2)180
n
(6 - 2)180
=
6
4 180
= 6
= 120
Measure of an angle =
#
Corollary to the Polygon Angle-Sum Theorem
Substitute 6 for n.
Simplify.
The measure of each interior angle in one hexagonal facet is 120.
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251
Problem 4
Using the Polygon Angle-Sum Theorem
How does the
diagram help you?
You know the number of
sides and four of the five
angle measures.
What is mjY in pentagon TODAY?
T
Use the Polygon Angle-Sum Theorem for n = 5.
m∠T + m∠O + m∠D + m∠A + m∠Y = (5 - 2)180
110 + 90 + 120 + 150 + m∠Y = 3
# 180
470 + m∠Y = 540
110
O
120 150
Substitute.
D
Simplify.
A
m∠Y = 70
Subtract 470 from each side.
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Problem 5
Investigating Exterior Angles of Polygons
Choose from a variety of tools (such as a ruler, a protractor, or a graphing
A calculator) to investigate exterior angles of polygons. Explain your choice.
A protractor is a useful tool for investigating exterior angles of polygons because
you use protractors to measure angles.
B Draw an exterior angle at each vertex of three different polygons. Investigate
patterns and write a conjecture about the exterior angles.
What polygons
can you draw to
investigate patterns?
If you draw a triangle,
a quadrilateral, and
a pentagon, you can
investigate patterns for
different numbers of
exterior angles in each
polygon.
Step 1Draw three different polygons. Then draw the exterior angles at each
vertex of the polygons as shown.
120°
135°
58°
63°
90°
79°
88°
105°
64°
130°
58°
90°
Step 2Use the protractor to measure the exterior angles of each polygon.
Observe any patterns. Write a conjecture about the exterior angles of
polygons.
Notice that for each polygon the sum of the measures of the exterior angles is 360.
Triangle: 135 + 120 + 105 = 360
Quadrilateral: 79 + 63 + 130 + 88 = 360
Pentagon:
90 + 90 + 58 + 58 + 64 = 360
Conjecture: The sum of the measures of the exterior angles of a polygon, one at
each vertex, is 360.
252
Y
Lesson 6-1 The Polygon Angle-Sum Theorems
Problem 6
TEKS Process Standard (1)(F)
Finding an Exterior Angle Measure
What is mj1 in the regular octagon at the right?
By the Polygon Exterior Angle-Sum Theorem, the sum of the exterior
angle measures is 360. Since the octagon is regular, the interior angles are
congruent. So their supplements, the exterior angles, are also congruent.
What kind of angle
is j1?
Looking at the diagram,
you know that ∠1 is an
exterior angle.
360
m∠1 = 8 NLINE
HO
ME
RK
O
= 45
WO
7
6
2
Divide 360 by 8, the number of sides in an octagon.
3
Simplify.
PRACTICE and APPLICATION EXERCISES
8
1
5
4
Scan page for a Virtual Nerd™ tutorial video.
hsm11gmse_0601_t06303
Find the measure of one interior angle in each regular polygon.
1. 2.
3.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
4.
Sketch an equilateral polygon that is not equiangular.
5.
A triangle has two congruent interior angles and an exterior angle that measures
100. Find two possible sets of interior angle measures for the triangle.
Analyze Mathematical Relationships (1)(F) Find the value of each variable.
6. y
110 z 100 87 7.
z
x
(z 13) w y (z 10)
8.
3x 2x 4x x
9.a. A polygon has n sides. An interior angle of the polygon and an adjacent
exterior
form a straight angle. What is the sum of the measures of the
hsm
11gm angle
se_0601_t06058.ai
hsm 11gm se_0601_t06060.ai
n straight angles? Of the nhsm
interior
11gmangles?
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b.Using your answers in part (a), what is the sum of the measures of the n
exterior angles? What theorem does this prove?
10.a. Use geometry software or other tool to explore the relationships among the
interior angles of quadrilaterals. Draw several quadrilaterals with parallel
opposite sides. Measure the interior angles.
b.Make two conjectures about the interior angles of this type of quadrilateral.
PearsonTEXAS.com
253
11.Explain Mathematical Ideas (1)(G) Your friend says she has
another way to find the sum of the interior angle measures of a
polygon. She picks a point inside the polygon, draws a segment to
each vertex, and counts the number of triangles. She multiplies
the total by 180, and then subtracts 360 from the product. Does
her method work? Explain.
12.The measure of an interior angle of a regular polygon is three times the measure
of an exterior angle of the same polygon. What is the name of the polygon?
hsm 11gm se_0601_t06061.ai
Apply Mathematics (1)(A) The gift package at the right contains fruit and cheese.
The fruit is in a container that has the shape of a regular octagon. The fruit
container fits in a square box. A triangular cheese wedge fills each
corner of the box.
13.Find the measure of each interior angle of a cheese wedge.
14.Display Mathematical Ideas (1)(G) Show how to rearrange the four pieces of
cheese to make a regular polygon. What is the measure of each interior angle of
the polygon?
15.a.Select Tools to Solve Problems (1)(C) Choose from a variety of tools (such
as a ruler, a compass, or geometry software) to investigate the exterior angles
of regular polygons. Explain your choice. Draw three regular polygons, each
with a different number of sides. Then draw the exterior angles at each vertex
of the polygons.
b.Make two conjectures about the exterior angles of regular polygons.
16.a.In the Corollary to the Polygon Angle-Sum Theorem, explain why the
measure of an interior angle of a regular n-gon is given by the formulas
180(n - 2)
and 180 - 360
n
n .
b.Use the second formula to explain what happens to the measures of the
interior angles of regular n-gons as n becomes a large number. Explain also
what happens to the polygons.
TEXAS Test Practice
Shr
ubs
17.The car at each vertex of a Ferris wheel holds a maximum of five people. The sum
of the interior angle measures of the Ferris wheel is 7740. What is the maximum
number of people the Ferris wheel can hold?
Maple Street
C
A
18.The Public Garden is located between two parallel streets:
64
Maple Street and Oak Street. The garden faces Maple Street
Public
Sh
and is bordered by rows of shrubs that intersect Oak Street at
Garden
rub
s
point B. What is m∠ABC, the angle formed by the shrubs?
19.△ABC ≅ △DEF . If m∠A = 3x + 4, m∠C = 2x, and
m∠E = 4x + 5, what is m∠B?
254
Lesson 6-1 The Polygon Angle-Sum Theorems
37
Oak Street
B
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6-2 Properties of Parallelograms
TEKS FOCUS
VOCABULARY
TEKS (6)(E) Prove a quadrilateral is a parallelogram,
rectangle, square, or rhombus using opposite sides,
opposite angles, or diagonals and apply these relationships
to solve problems.
•Consecutive angles – Consecutive angles of a polygon
TEKS (1)(F) Analyze mathematical relationships to
connect and communicate mathematical ideas.
•Opposite sides – Opposite sides of a quadrilateral are
Additional TEKS (1)(G)
•Parallelogram – A parallelogram is a quadrilateral with
share a common side.
•Opposite angles – Opposite angles of a quadrilateral are
two angles that do not share a side.
two sides that do not share a vertex.
two pairs of parallel sides.
•Analyze – closely examine objects, ideas, or relationships
to learn more about their nature
ESSENTIAL UNDERSTANDING
Parallelograms have special properties regarding their sides, angles, and diagonals.
Key Concept Parallelograms and Their Parts
Term Description
Diagram
A parallelogram is a quadrilateral with both
pairs of opposite sides parallel.
You can abbreviate parallelogram with the
symbol ▱.
In a quadrilateral, opposite sides do not
share a vertex and opposite angles do not
share a side.
B
C
AB
and
CD
hsm11gmse_0602_t06469.ai
are opposite
sides.
A
D
Angles of a polygon that share a side are
consecutive angles. In the diagram, ∠A and
∠B are consecutive angles because they
share side AB.
A
A and C
are opposite
angles.
B
B and C
are also
hsm11gmse_0602_t06471.ai
consecutive
angles.
C
D
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PearsonTEXAS.com
255
Theorem 6-3
Theorem
If a quadrilateral is a
parallelogram, then its opposite
sides are congruent.
If . . .
ABCD is a ▱
B
C
Then . . .
AB ≅ CD and BC ≅ DA
B
C
D
A
A
D
For a proof of Theorem 6-3, see the Reference section on page 683.
Theorem 6-4
Theorem
If a quadrilateral is a
parallelogram, then its
consecutive angles are
supplementary.
hsm11gmse_0603_t06433.ai hsm11gmse_0602_t06473.ai
If . . .
ABCD is a ▱
B
C
Then . . .
B
C m∠A + m∠B = 180
m∠B + m∠C = 180
D
A
m∠C
+ m∠D = 180
m∠D + m∠A = 180
D
A
You will prove Theorem 6-4 in Exercise 21.
Theorem 6-5
Theorem
If a quadrilateral is a
parallelogram, then its opposite
angles are congruent.
hsm11gmse_0603_t06432.ai
hsm11gmse_0603_t06433.ai
If . . .
Then . . .
ABCD is a ▱.
∠A ≅ ∠C and ∠B ≅ ∠D
B
B
C
C
A
D
A
D
For a proof of Theorem 6-5, see Problem 2.
Theorem 6-6
Theorem
If a quadrilateral is a
parallelogram, then its
diagonals bisect each other.
hsm11gmse_0603_t06434.ai
hsm11gmse_0602_t06487.ai
If
...
Then . . .
ABCD is a ▱
AE ≅ CE and BE ≅ DE
B
C
B
C
D
A
A
D
E
You will prove Theorem 6-6 in Exercise 11.
Theorem 6-7
Theorem
If three (or more) parallel lines
cut off congruent segments
on one transversal, then they
cut off congruent segments on
every transversal.
hsm11gmse_0603_t06433.ai
If
< .> . .< > < >
AB } CD } EF and AC ≅ CE
A
C
E
hsm11gmse_0603_t06443.ai
Then . . .
BD ≅ DF
A
B
C
D
F
E
B
D
F
You will prove Theorem 6-7 in Exercise 23.
256
Lesson 6-2 Properties of Parallelograms
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Problem 1
Using Consecutive Angles
Q
Multiple Choice What is mjP in ▱PQRS?
What information
from the diagram
helps you get
started?
From the diagram, you
know m∠PSR and that
∠P and ∠PSR are
consecutive angles. So you
can write an equation
and solve for m∠P.
26
116
64
126
P
R
64
m∠P + m∠S = 180
Consecutive angles of a ▱ are
supplementary.
m∠P + 64 = 180
S
Substitute.
m∠P = 116
Subtract 64 from each side.
The correct answer is C.
Problem 2
Proof
TEKS Process Standard (1)(G)
Using Properties of Parallelograms in a Proof
Given: ▱ABCD
B
C
Prove: ∠A ≅ ∠C and ∠B ≅ ∠D
A
D
ABCD is a ▱.
Given
Why is a flow proof
useful here?
A flow proof allows you
to see how the pairing of
two statements leads to
a conclusion.
hsm11gmse_0602_t06478.ai
∠A and ∠B are
consecutive ⦞.
∠B and ∠C are
consecutive ⦞.
∠C and ∠D are
consecutive ⦞.
Def. of consecutive ⦞
Def. of consecutive ⦞
Def. of consecutive ⦞
∠A and ∠B are
supplementary.
∠B and ∠C are
supplementary.
∠C and ∠D are
supplementary.
Consecutive ⦞
are supplementary.
Consecutive ⦞
Consecutive ⦞
are supplementary.
are supplementary.
∠A ≅ ∠C
∠B ≅ ∠D
Supplements of the
same ∠ are ≅.
Supplements of the
same ∠ are ≅.
hsm11gmse_0602_t06480.ai
PearsonTEXAS.com
257
Problem 3
Using Algebra to Find Lengths
L
Solve a system of linear equations to find the values of x and y
in ▱KLMN . What are KM and LN?
y
x
10
K
The diagonals of a
parallelogram bisect each
other.
M
8
2x y
P 2
N
hsm11gmse_0602_t06481.ai
KP ≅ M P
LP ≅ N P
① y + 10 = 2x − 8
② x=y+2
Set up a system of linear
equations by substituting
the algebraic expressions
for each segment length.
Substitute ( y + 2) for x
in equation ①. Then solve
for y.
y + 10 = 2(y + 2) − 8
y + 10 = 2y + 4 − 8
y + 10 = 2y − 4
10 = y − 4
14 = y
Substitute 14 for y in
equation ②. Then solve
for x.
x = 14 + 2
= 16
Use the values of x and y
to find KM and LN.
KM = 2(KP)
= 2(y + 10) = 2(14 + 10)
= 48
LN = 2(LP)
= 2(x)
= 2(16)
= 32
Problem 4
TEKS Process Standard (1)(F)
Using Parallel Lines and Transversals
< > < > < > < >
In the figure at the right, AE } BF } CG } DH ,
What information do
you need?
You know the length of
EF. To find EH, you need
the lengths of FG and
GH.
AB = BC = CD = 2, and EF = 2.25. What is EH?
Since } lines divide AD
into equal parts, they also
divide EH into equal parts.
EF = FG = GH
EH = EF + FG + GH
EH = 2.25 + 2.25 + 2.25 = 6.75
Segment Addition Postulate
A
B
C
D
E
F
G
H
Substitute.
hsm11gmse_0602_t06485.ai
258
Lesson 6-2 Properties of Parallelograms
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
1.
What are the values of x and y in the parallelogram?
For additional support when
completing your homework,
go to PearsonTEXAS.com.
y
2.
The perimeter of ▱ABCD is 92 cm. AD is 7 cm more
than twice AB. Find the lengths of all four sides of ▱ABCD.
In the figure, PQ = QR = RS. Find each length.
3.
ZU
4.XZ
5.
TU
6.XV
7.
YX
8.YV
9.
WX
10.WV
3y W
3x S
U
2.25
Y
hsm11gmse_0602_t06078.ai
Z R
3
X
T
11.Justify Mathematical Arguments (1)(G) Proof Complete this two-column proof of Theorem 6-6.
Q
P
V
B
Given: ▱ABCD
2
4
C
E
1
3
hsm11gmse_0602_t06077.ai
A
D
Reasons
Prove: AC and BD bisect each other at E.
Statements
1) ABCD is a parallelogram.
1) Given
2) AB } DC
2) a. ?
3) ∠1 ≅ ∠4; ∠2 ≅ ∠3
3) b. ?
4) AB ≅ DC
4) c. ?
5) d. 5) ASA
?
hsm11gmse_0602_t06075.ai
6) AE ≅ CE; BE ≅ DE
6) e. 7) f. 7) Definition of bisector
?
Find the values of x and y in ▱PQRS.
?
Q
R
12.PT = 2x, TR = y + 4, QT = x + 2, TS = y
T
13.PT = x + 2, TR = y, QT = 2x, TS = y + 3
P
14.PT = y, TR = x + 3, QT = 2y, TS = 3x - 1
Use the diagram at the right for each proof.
S
S
Y
T
15.Given: ▱RSTW and ▱XYTZ
Proof
hsm11gmse_0602_t06076.ai
X
Z
R
W
Prove: ∠R ≅ ∠X
16.Given: ▱RSTW and ▱XYTZ
Proof
Prove: XY } RS
Find the measures of the numbered angles for each parallelogram.
17.18.
19.
28
3
3
1
38
2
81
110
1
2
hsm11gmse_0602_t06087.ai
3
85
1
48
2
PearsonTEXAS.com 259
hsm11gmse_0602_t06132.ai
hsm11gmse_0602_t06131.ai
hsm11gmse_0602_t06134.ai
20.Apply Mathematics (1)(A) A pantograph is an expandable
device, shown at the right. Pantographs are used in the
television industry in positioning lighting and other equipment.
In the photo, points D, E, F, and G are the vertices of a
parallelogram. ▱DEFG is one of many parallelograms that
change shape as the pantograph extends and retracts.
a.If DE = 2.5 ft, what is FG?
b.If m∠E = 129, what is m∠G?
c.What happens to m∠D as m∠E increases or decreases?
Explain.
21.Prove Theorem 6-4.
B
Proof
E
D
F
G
C
Given: ▱ABCD
Prove: ∠A is supplementary to ∠B.
∠A is supplementary to ∠D.
A
D
22.Explain Mathematical Ideas (1)(G) Is there an SSSS
congruence theorem for parallelograms? Explain.
23.Prove Theorem 6-7. Use the diagram at the hsm11gmse_0602_t06084.ai
right.
B
A
< > < > < >
Proof
Given: AB } CD } EF , AC ≅ CE
3
D
C 1
Prove: BD ≅ DF
G 2
6
F
E 4
24.Explain Mathematical Ideas (1)(G) Explain how to separate
H
5
a blank card into three strips that are the same height by
using lined paper, a straightedge, and Theorem 6-7.
TEXAS Test Practice
hsm11gmse_0602_t06136.ai
P
25.PQRS is a parallelogram with m∠Q = 4x and m∠R = x + 10. Which statement
explains why you can use the equation 4x + (x + 10) = 180 to solve for x?
Q
A.The measures of the interior angles of a quadrilateral have a sum of 360.
B.Opposite sides of a parallelogram are congruent.
C.Opposite angles of a parallelogram are congruent.
D.Consecutive angles of a parallelogram are supplementary.
S
26.In the figure of DEFG at the right, DE } GF . Which statement must be true?
F.
m∠D + m∠E = 180
H.DE ≅ GF
G.m∠D + m∠G = 180
J.DG ≅ EF
R
D
E
hsm11gmse_0602_t12794
G
F
27.An obtuse triangle has side lengths of 5 cm, 9 cm, and 12 cm. What is the length of
the side opposite the obtuse angle?
A.5 cm
D.not enough information
hsm11gmse_0602_t06139.ai
28.Find the measure of one exterior angle of a regular hexagon. Explain your method.
260
B.9 cm
Lesson 6-2 Properties of Parallelograms
C.12 cm
6-3 Proving That a Quadrilateral
Is a Parallelogram
TEKS FOCUS
VOCABULARY
TEKS (6)(E) Prove a quadrilateral is a parallelogram, rectangle, square,
or rhombus using opposite sides, opposite angles, or diagonals and apply
these relationships to solve problems.
•Analyze – closely examine objects, ideas,
or relationships to learn more about their
nature
TEKS (1)(F) Analyze mathematical relationships to connect and
communicate mathematical ideas.
Additional TEKS (1)(G)
ESSENTIAL UNDERSTANDING
You can decide whether a quadrilateral is a parallelogram if its sides,
angles, and diagonals have certain properties.
Theorem 6-8
Theorem
If . . .
If both pairs of opposite sides
of a quadrilateral are congruent,
then the quadrilateral is a
parallelogram.
B
A
Then . . .
C
ABCD is a ▱
B
C
AB ≅ CD
BC ≅ DA
D
A
D
For a proof of Theorem 6-8, see Problem 1.
Theorem 6-9
Theorem
If an angle of a quadrilateral is
supplementary to both of its
consecutive angles, then the
quadrilateral is a parallelogram.
hsm11gmse_0602_t06473.ai
If . . .
B
A
C
D
m∠A + m∠B = 180
m∠A + m∠D = 180
hsm11gmse_0603_t06433.ai
Then . . .
ABCD is a ▱
B
C
A
D
You will prove Theorem 6-9 in Exercise 17.
Theorem 6-10
Theorem
If both pairs of opposite angles
of a quadrilateral are congruent,
then the quadrilateral is a
parallelogram.
hsm11gmse_0603_t06432.ai
If . . .
B
A
C
∠A ≅ ∠C
∠B ≅ ∠D
D
hsm11gmse_0603_t06433.ai
Then . . .
ABCD is a ▱
B
C
A
D
For a proof of Theorem 6-10, see Problem 2.
hsm11gmse_0603_t06434.ai
hsm11gmse_0603_t06433.ai
PearsonTEXAS.com
261
Theorem 6-11
Theorem
If the diagonals of a
quadrilateral bisect each other,
then the quadrilateral is a
parallelogram.
If . . .
B
A
E
Then . . .
C
D
ABCD is a ▱
B
C
AE ≅ CE
BE ≅ DE
A
D
For a proof of Theorem 6-11, see Problem 3.
hsm11gmse_0603_t06443.ai
Theorem 6-12
Theorem
If one pair of opposite sides
of a quadrilateral is both
congruent and parallel, then the
quadrilateral is a parallelogram.
hsm11gmse_0603_t06433.ai
If . . .
Then . . .
B
A
ABCD is a ▱
B
C
C
D
BC ≅ DA
BC } DA
A
D
You will prove Theorem 6-12 in Exercise 16.
hsm11gmse_0603_t06448.ai hsm11gmse_0603_t06433.ai
Concept Summary P
roving That a Quadrilateral Is
a Parallelogram
Method
Source
Prove that both pairs of opposite sides are parallel.
Definition of
parallelogram
Prove that both pairs of opposite sides are congruent.
Theorem 6-8
Diagram
hsm11gmse_0603_t06461.ai
Prove that an angle is supplementary to both of its
consecutive angles.
Theorem 6-9
75
75 105
hsm11gmse_0603_t06463.ai
Prove that both pairs of opposite angles are
congruent.
Theorem 6-10
hsm11gmse_0603_t12047
Prove that the diagonals bisect each other.
Theorem 6-11
hsm11gmse_0603_t06465.ai
Prove that one pair of opposite sides is congruent
and parallel.
262
Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram
Theorem 6-12
hsm11gmse_0603_t06467.ai
hsm11gmse_0603_t06468.ai
Problem 1
B
Proof Proving Theorem 6-8
C
Given: AB ≅ CD, BC ≅ DA
Prove: ABCD is a parallelogram.
Why do you start by
drawing BD?
In many proofs about
parallelograms, it is
convenient to have a pair
of triangles that you can
show to be congruent.
Draw a diagonal to form
two triangles.
Statements
A
D
Reasons
1) Draw BD.
1) Construction
2) AB ≅ CD and BC ≅ DA
2) Given
hsm11gmse_0603_t06430.ai
3) BD ≅ BD
3) Reflexive Property of Congruence
4) △ABD ≅ △CDB
4) SSS
5) ∠ADB ≅ ∠CBD and
∠CDB ≅ ∠ABD
5) Corresponding parts of congruent
triangles are congruent.
6) AB } DC and BC } AD
6) Converse of the Alternate Interior
Angles Theorem
7) ABCD is a parallelogram.
7) Definition of parallelogram
Problem 2
TEKS Process Standard (1)(G)
B
Proof Proving Theorem 6-10
How do you get
started with the
proof?
Since the goal is to show
that opposite sides are
parallel, you can label the
angle measures as in the
diagram and show that
same-side interior angles
are supplementary.
x
Given: ∠A ≅ ∠C, ∠B ≅ ∠D
Prove: ABCD is a parallelogram.
Statements
C
A
y
D
Reasons
1) ∠A ≅ ∠C, ∠B ≅ ∠D
1) Given
2) x + y + x + y = 360
2) The
sum of the measures of the
hsm11gmse_0603_t06158.ai
angles of a quadrilateral is 360.
3) 2(x + y) = 360
3) Distributive Property
4) x + y = 180
4) Division Property of Equality
5) ∠A and ∠B are supplementary.
∠A and ∠D are supplementary.
5) Definition of supplementary angles
6) AD } BC, AB } DC
6) Converse of the Same-Side Interior
Angles Postulate
7) ABCD is a parallelogram.
7) Definition of parallelogram
PearsonTEXAS.com
263
Problem 3
Proof Proving Theorem 6-11
How can you get
started?
Notice that in the
diagram there are several
pairs of triangles. Use
the given information to
prove pairs of triangles
congruent. Then use their
corresponding parts to
show that ABCD is a
parallelogram.
B
C
Given: AC and BD bisect each other at E.
E
Prove: ABCD is a parallelogram.
A
D
AC and BD bisect each other at E.
Given
hsm11gmse_0603_t06445.ai
∠AEB ≅ ∠CED
AE ≅ CE
BE ≅ DE
∠BEC ≅ ∠DEA
Vertical ⦞ are ≅.
Def. of segment bisector
Vertical ⦞ are ≅.
△AEB ≅ △CED
△BEC ≅ △DEA
SAS
SAS
∠BAE ≅ ∠DCE
∠ECB ≅ ∠EAD
Corresp. parts of ≅
are ≅.
Corresp. parts of ≅
are ≅.
AB CD
BC AD
If alternate interior ⦞ ≅,
then lines are .
If alternate interior ⦞ ≅,
then lines are .
ABCD is a parallelogram.
Def. of parallelogram
Problem 4
Finding Values for Parallelograms
Which theorem
should you use?
The diagram gives you
information about sides.
Use Theorem 6-8 because
it uses sides to conclude
that a quadrilateral is a
parallelogram.
A For what value of y must PQRS be a parallelogram?
3x 5
P
Q
Step 1Find x.
If opp. sides are ≅ , then the quad.
3x - 5 = 2x + 1
is a ▱ .
x-5=1
x=6
Subtract 2x from each side.
Add 5 to each side.
Step 2Find y.
y
x2
S
2x 1
R
8
. . . . . . .
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0
1
2
3
4
5
6
7
8
9
0 0 0
1 1 1
2 2 2
3 3 3
4 4 4
5 5 5
6 6 6
7 7 7
8 8 8
9 9 9
hsm11gmse_0603_t06439.ai
y = x + 2
If opp. sides are ≅ , then the quad. is a ▱ .
= 6 + 2 Substitute 6 for x.
=8
Simplify.
For PQRS to be a parallelogram, the value of y must be 8.
continued on next page ▶
264
Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram
Problem 4
continued
B For what values of w and z must ABCD be a parallelogram?
(7z 1 5)8
A
Step 1Find w.
5w - 30 = 3w + 10 If opp. angles are ≅, then the quad. is a ▱ .
2w - 30 = 10
2w = 40
Add 30 to each side.
w = 20
Divide each side by 2.
Subtract 3w from each side.
(5w 2 30)8
(3w 1 10)8
D
B
C
(8z 2 10)8
Step 2Find z.
8z - 10 = 7z + 5 If opp. angles are ≅, then the quad. is a ▱ .
z - 10 = 5
Subtract 7z from each side.
z = 15
Add 10 to each side.
For ABCD to be a parallelogram, the value of w must be 20 and the value
of z must be 15.
Problem 5
TEKS Process Standard (1)(F)
Deciding Whether a Quadrilateral Is a Parallelogram
How do you decide
if you have enough
information?
If you can satisfy every
condition of a theorem
about parallelograms,
then you have enough
information.
Can you prove that the quadrilateral is a parallelogram based on the given
information? If so, write a paragraph proof. If not, explain.
A Given: AB = 5, CD = 5,
B Given: HI ≅ HK, JI ≅ JK
m∠A = 50, m∠D = 130
Prove: HIJK is a parallelogram.
Prove: ABCD is a parallelogram.
A
5
H
I
B
50
130
D
5
C
Yes. Proof: Because it is given that
m∠A = 50 and m∠D = 130,
same-side interior angles A and D
hsm11gmse_0603_t06451.ai
are supplementary.
So AB } CD.
It is given that AB = 5 and CD = 5,
so AB ≅ CD. Therefore, ABCD is a
parallelogram by Theorem 6-12.
K
J
No. By Theorem 6-8, you need to
show that both pairs of opposite
sides, not consecutive sides, are
hsm11gmse_0603_t06453.ai
congruent.
continued on next page ▶
PearsonTEXAS.com
265
Problem 5
continued
C Given: m∠N = m∠Q = 39,
D Given: GE = 24, GH = 12,
m∠P = 141
DF = 32, HF = 16
Prove: MNPQ is a parallelogram.
M
Prove: DEFG is a parallelogram.
N
D
398
398
P
Q
E
H
1418
G
Yes. Proof: It is given that
m∠N = m∠Q = 39 and
m∠P = 141. Since the sum of the
angle measures of a quadrilateral is
360, m∠M = 141. Since m∠N = m∠Q
and m∠P = m∠M, ∠N ≅ ∠Q and
∠M ≅ ∠P. Therefore, MNPQ is a
parallelogram by Theorem 6-10.
16
12
F
Yes. Proof: It is given that GE = 24,
GH = 12, DF = 32, and HF = 16. By
the Segment Addition Postulate,
HE = 12 and DH = 16, so the
diagonals of the quadrilateral bisect
each other. DEFG is a parallelogram
by Theorem 6-11.
Problem 6
Identifying Parallelograms
As the arms of the lift
move, what changes
and what stays the
same?
The angles the arms form
with the ground and the
platform change, but the
lengths of the arms and
the platform stay the
same.
Vehicle Lifts A truck sits on the platform of a vehicle lift. Two moving arms
raise the platform until a mechanic can fit underneath. Why will the truck
always remain parallel to the ground as it is lifted? Explain.
Q
Q
R
R
26 ft
6 ft
P
26 ft
6 ft
6 ft
26 ft
S
6 ft
P
26 ft
The angles of PQRS change as platform QR rises, but its side lengths remain the
same. Both pairs of opposite sides are congruent, so PQRS is a parallelogram by
Theorem 6-8. By the definition of a parallelogram, PS } QR. Since the base of the
lift PS lies along the ground, platform QR, and therefore the truck, will always be
parallel to the ground.
266
Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram
S
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
1.Given: AB ≅ CD, DE ≅ FC, EA ≅ BF 2.Given: ∠M ≅ ∠P, ∠MNQ ≅ ∠PQN,
Proof
Proof
∠MQN ≅ ∠PNQ
Prove: ABCD is a parallelogram.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
A
Prove: MNPQ is a parallelogram.
B
M
E
N
F
D
C
Q
P
3.Given: M is the midpoint of HK and JL.
4.Given: ∠A and ∠C are right angles,
Proof
Proof
AD ≅ CB
Prove: HJKL is a parallelogram.
H
Prove: ABCD is a parallelogram.
J
M
A
L
B
K
D
C
Analyze Mathematical Relationships (1)(F) For what values of x and y must
ABCD be a parallelogram?
5.
B
(y 78)
3y 2x
(4x 21)
A
8.
A
6. C
3x D
D
2y
1
4
y
A
A
2y 2
B
5y
D
D
3y 9
B
C
(2x 15)
hsm11gmse_0603_t06152.ai
(4x 33)
6
3x
y4
hsm11gmse_0603_t06149.ai
C
C
2x 7
10. B
(3x 10)
hsm11gmse_0603_t06150.ai
(8x 5)
5x 8
A
B
9.
B
7. D
C
7
A
C
D
11.Display Mathematical Ideas (1)(G) Sketch two noncongruent parallelograms
ABCD and EFGH so that AB ≅ EF and BC ≅ FG.
hsm11gmse_0603_t06162.ai
hsm11gmse_0603_t06160.ai
hsm11gmse_0603_t06161.ai
Can you prove that the quadrilateral is a parallelogram based on the given
information? Explain.
12.
hsm11gmse_0603_t06154.ai
13. hsm11gmse_0603_t06155.ai
14.
PearsonTEXAS.com
267
hsm11gmse_0603_t06157.ai
15.Apply Mathematics (1)(A) Quadrilaterals are
formed on the side of this fishing tackle box
by the adjustable shelves and connecting pieces.
Explain why the shelves are always parallel to
each other no matter what their position is.
A
B
D
C
16.Justify Mathematical 17.Prove Theorem 6-9.
Proof Arguments (1)(G) Prove
Proof
Given: ∠A is supplementary to ∠B.
Theorem 6-12.
∠A is supplementary to ∠D.
Given: BC } DA, BC ≅ DA
Prove: ABCD is a parallelogram.
Prove: ABCD is a parallelogram.
B
A
TEXAS Test Practice
B
C
C
A
D
D
hsm11gmse_0603_t06581.ai
hsm11gmse_0603_t06159.ai
18.Which piece of additional information would allow you
to prove that PQRS is a parallelogram?
A.PQ ≅ RS
C.∠PTQ ≅ ∠RTS
B.QR ≅ SP
D.∠QPR ≅ ∠SRP
P
Q
T
S
R
19.In quadrilateral ABCD, m∠A = 3x + 2, m∠B = x - 22, and m∠C = 2x + 52.
Which value of x allows you to conclude that ABCD is a parallelogram?
F.
50
G.34
H.28
J. -12
20.Quadrilateral JKLM is a parallelogram. Which of the following
does NOT guarantee that JNPM is a parallelogram?
N
J
K
A.N is the midpoint of JK and P is the midpoint of ML.
B.JM ≅ NP
M
L
P
C.JM } NP
D.∠JMP ≅ ∠NPL
21.Write a proof using the diagram.
N
Prove: JNTC is a parallelogram.
268
Lesson 6-3 Proving That a Quadrilateral Is a Parallelogram
T
P
Given: △NRJ ≅ △CPT, JN } CT
J
R
C
hsm11gmse_0603_t06167.ai
6-4 Properties of Rhombuses, Rectangles,
and Squares
TEKS FOCUS
VOCABULARY
•Rectangle – A rectangle is a parallelogram with
TEKS (5)(A) Investigate patterns to make conjectures about
geometric relationships, including angles formed by parallel lines
cut by a transversal, criteria required for triangle congruence,
special segments of triangles, diagonals of quadrilaterals, interior
and exterior angles of polygons, and special segments and angles
of circles choosing from a variety of tools.
four right angles.
•Rhombus – A rhombus is a parallelogram with four
congruent sides.
•Square – A square is a parallelogram with four
congruent sides and four right angles.
TEKS (1)(C) Select tools, including real objects, manipulatives,
paper and pencil, and technology as appropriate, and techniques,
including mental math, estimation, and number sense as
appropriate, to solve problems.
•Number sense – the understanding of what
numbers mean and how they are related
Additional TEKS (1)(F), (6)(E)
ESSENTIAL UNDERSTANDING
The parallelograms in the Take Note box below have basic properties about their sides
and angles that help identify them. The diagonals of these parallelograms also have
certain properties.
Key Concept Special Parallelograms
A rhombus is a parallelogram
with four congruent sides.
A rectangle is a parallelogram
with four right angles.
A square is a parallelogram
with four congruent sides and
four right angles.
hsm11gmse_0604_t06019
Theorem 6-13
hsm11gmse_0604_t06018
hsm11gmse_0604_t06020
Theorem
If a parallelogram is a
rhombus, then its diagonals
are perpendicular.
If . . .
ABCD is a rhombus
A
D
Then . . .
AC # BD
B
B
C
A
D
C
For a proof of Theorem 6-13, see Lesson 7-3.
hsm11gmse_0604_t06022
PearsonTEXAS.com 269
hsm11gmse_0604_t06023
Theorem 6-14
Theorem
If a parallelogram is a rhombus,
then each diagonal bisects a
pair of opposite angles.
If . . .
ABCD is a rhombus
A
D
B
C
Then . . .
A
D
2
3
1
4
7
5
8
6
B
C
∠1 ≅ ∠2
∠3 ≅ ∠4
∠5 ≅ ∠6
∠7 ≅ ∠8
You will prove Theorem 6-14 in Exercise 10.
hsm11gmse_0604_t06024
Theorem
Theorem
If a parallelogram is a
rectangle, then its
diagonals are congruent.
6-15 hsm11gmse_0604_t06022
If . . .
ABCD is a rectangle
D
A
B
C
Then . . .
AC ≅ BD
A
D
B
C
You will prove Theorem 6-15 in Exercise 13.
Problem 1
hsm11gmse_0604_t06028
hsm11gmse_0604_t06031
Classifying Special Parallelograms
How do you decide
whether ABCD is a
rhombus, a rectangle,
or a square?
Use the definitions of
rhombus, rectangle, and
square along with the
markings on the figure.
270
Is ▱ABCD a rhombus, a rectangle, or a square?
Explain.
▱ABCD is a rectangle. Opposite angles of a
parallelogram are congruent, so m∠D is 90. By the
Same-Side Interior Angles Theorem, m∠A = 90
and m∠C = 90. Since ▱ABCD has four right
angles, it is a rectangle. You cannot conclude that
ABCD is a square because you do not know its
side lengths.
Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares
A
B
E
F
H
D
G
C
Problem 2
TEKS Process Standards (1)(C)
Investigating Diagonals of Quadrilaterals
Choose from a variety of tools (such as a protractor, a ruler, a compass,
A or a geoboard) to investigate patterns in the diagonals of quadrilaterals.
Explain your choice.
A manipulative such as a geoboard makes it easy to make different types of
quadrilaterals and their diagonals.
Make several parallelograms, rectangles, and rhombuses. Then make a
B conjecture about the diagonals of each type of quadrilateral.
Parallelogram
Rectangle
Rhombus
Conjecture: The diagonals
of a parallelogram bisect
each other.
Conjecture: The diagonals
of a rectangle are congruent.
Conjecture: The diagonals
of a rhombus are
perpendicular.
How can you measure
distances on a
geoboard?
You can use the grid
of pegs to indicate
horizontal and vertical
units.
Problem 3
How are the
numbered angles
formed?
The angles are formed
by diagonals. Use what
you know about the
diagonals of a rhombus
to find the angle
measures.
TEKS Process Standard (1)(F)
Finding Angle Measures
What are the measures of the numbered angles in rhombus ABCD?
m∠1 = 90 The diagonals of a rhombus are #.
m∠2 = 58 Alternate Interior Angles Theorem
m∠3 = 58
m∠1 + m∠3 + m∠4 = 180
90 + 58 + m∠4 = 180
148 + m∠4 = 180
m∠4 = 32 E ach diagonal of a rhombus bisects a
pair of opposite angles.
B
C
58
A
4
1 2
3
D
Triangle Angle-Sum Theorem
Substitute.
Simplify.
hsm11gmse_0604_t06026
Subtract 148 from each side.
PearsonTEXAS.com
271
Problem 4
Finding Diagonal Length
Multiple Choice In rectangle RSBF, SF = 2x + 15 and RB = 5x − 12.
What is the length of a diagonal?
How can you find the
length of a diagonal?
Since RSBF is a rectangle
and its diagonals are
congruent, use the
expressions to write an
equation.
1
9
18
You know that the diagonals of
a rectangle are congruent, so
their lengths are equal.
RK
O
HO
WO
R
F
33
hsm11gmse_0604_t06032
2x + 15 = 5x − 12
15 = 3x − 12
27 = 3x
9=x
RB = 5x − 12
= 5(9) − 12
= 33
The correct answer is D.
Substitute 9 for x in the
expression for RB.
ME
B
SF = RB
Set the algebraic expressions for
SF and RB equal to each other
and find the value of x.
NLINE
S
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
Find the measures of the numbered angles in each rhombus.
1.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
3
2.
4
2
35 1
3
3.
2
1
3
1
2
35
60
LMNP is a rectangle. Find the value of x and the length of each diagonal.
4.
LN = x and MP = 2x - 4
5.LN = 5x - 8 and MP = 2x + 1
hsm 11gm se_0604_t05921.aihsm 11gm se_0604_t05935.ai
hsm
11gm
se_0604_t05920.ai
6.
LN = 3x + 1 and MP = 8x - 4
7.LN = 9x - 14 and MP = 7x + 4
8.
LN = 7x - 2 and MP = 4x + 3
9.LN = 3x + 5 and MP = 9x - 10
A
10.Prove Theorem 6-14.
Proof
3
Given: ABCD is a rhombus.
Prove: AC bisects ∠BAD and ∠BCD.
B
272
Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares
4
D
2 1
C
hsm11gmse_0604_t06250.ai
Decide whether the parallelogram is a rhombus, a rectangle, or a square. Explain.
11.
12.
13.Justify Mathematical Arguments (1)(G) Complete the flow
Proof proof of Theorem 6-15.
A
D
Given: ABCD is a rectangle.
B
C
Prove: AC ≅ BD
ABCD is a ▱.
e.
b.
ABCD is
a rectangle.
Opposite sides
hsmof11gm se_0604_t06239.ai
a ▱ are ≅.
f.
BC ≅ BC
a.
AC ≅ BD
SAS
c.
∠ABC and ∠DCB
are right ⦞.
h.
∠ABC ≅ ∠DCB
g.
d.
14.Connect Mathematical Ideas (1)(F) Summarize the properties of squares that
follow from a square being (a) a parallelogram,
(b) a rhombus, and (c) a rectangle.
hsm11gmse_0604_t06243.ai
K 4b 6r J
15.
Analyze Mathematical Relationships (1)(F) Find the angle
measures and the side lengths of the rhombus at the right.
16.Create Representations to Communicate Mathematical Ideas
(1)(E) On graph paper, draw a parallelogram that is neither a
rectangle nor a rhombus.
r1
H
x
2r 4 G
b3
(2x 6)
ABCD is a rectangle. Find the length of each diagonal.
17.AC = 2(x - 3) and BD = x + 5
3y
19.AC = 5 and BD = 3y - 4
18. AC = 2(5a + 1) and BD = 2(a + 1)
3c
20. AC = 9 and BD = 4 - c
hsm11gmse_0604_t06254.ai
PearsonTEXAS.com
273
Find the values of the variables. Then find the side lengths.
21.rhombus 22.square 15
3y
2x 7
y1
5x
4x 3
2y 5
3y 9
23.Justify Mathematical Arguments (1)(G) Write a proof.
P
L
Proof
Given: Rectangle PLAN
hsm 11gm se_0604_t05942.ai
Prove: △LTP ≅ △NTA
T
hsm 11gm se_0604_t05943.ai
N
A
24.a. Select Tools to Solve Problems (1)(C) To investigate the diagonals and the
interior angles of rhombuses, choose from the following tools: ruler, paper
folding, or graphing calculator. Explain your choice.
b.Make several rhombuses with their diagonals. Observe anyhsm11gmse_0604_t06262.ai
patterns. Make a
conjecture about the diagonals and the interior angles of rhombuses.
Find the value of x in the rhombus.
25. 2
(7x 10)
(6x 2 3x)
26.
(2x 2 25x)
(3x 2 60)
TEXAS Test Practice
hsm 11gm se_0604_t05944.ai
hsm 11gm se_0604_t05945.ai
27.A part of a design for a quilting pattern consists of a regular pentagon
and five isosceles triangles, as shown. What is m∠1?
A.18
C.72
B.36
D.108
1
28.Which statement is true for some, but not all, rectangles?
F.
Opposite sides are parallel.
G.It is a parallelogram.
H.Adjacent sides are perpendicular.
J.All sides are congruent.
29.Which term best describes AD in △ABC?
A.altitude
C.median
B.angle bisector
D.perpendicular bisector
A
hsm11gmse_0604_t12846
B
D
C
30.Write the first step of an indirect proof that △PQR is not a right triangle.
hsm 11gm se_0604_t05947.ai
274
Lesson 6-4 Properties of Rhombuses, Rectangles, and Squares
6-5 Conditions for Rhombuses, Rectangles,
and Squares
TEKS FOCUS
VOCABULARY
•Analyze – closely examine objects, ideas, or relationships
TEKS (6)(E) Prove a quadrilateral is a parallelogram,
rectangle, square, or rhombus using opposite sides,
opposite angles, or diagonals and apply these relationships
to solve problems.
to learn more about their nature
TEKS (1)(F) Analyze mathematical relationships to
connect and communicate mathematical ideas.
Additional TEKS (1)(G)
ESSENTIAL UNDERSTANDING
You can determine whether a parallelogram is a rhombus or a rectangle based on the
properties of its diagonals.
Theorem 6-16
Theorem
If a quadrilateral is a
parallelogram with
perpendicular diagonals,
then the quadrilateral is a
rhombus.
If . . .
ABCD is a ▱ and AC # BD
A
D
Then . . .
ABCD is a rhombus
A
D
B
B
C
C
For a proof of Theorem 6-16, see Problem 1.
hsm11gmse_0605_t06034.ai
Theorem 6-17
Theorem
If a quadrilateral is a
parallelogram with a
diagonal that bisects a pair
of opposite angles, then the
quadrilateral is a rhombus.
If . . .
ABCD is a ▱, ∠1 ≅ ∠2, and
∠3 ≅ ∠4
A
D
3
4
1
B
hsm11gmse_0605_t06274.ai
Then . . .
ABCD is a rhombus
A
D
B
2
C
C
You will prove Theorem 6-17 in Exercise 16.
hsm11gmse_0605_t06274.ai
hsm11gmse_0605_t06036.ai
PearsonTEXAS.com
275
Theorem 6-18
Theorem
If a quadrilateral is a
parallelogram with congruent
diagonals, then the
quadrilateral is a rectangle.
If . . .
ABCD is a ▱, and AC ≅ BD
A
D
C
B
Then . . .
ABCD is a rectangle
D
A
C
B
You will prove Theorem 6-18 in Exercise 17.
hsm11gmse_0605_t06037.ai
Theorem 6-19
Theorem
If a quadrilateral is a
parallelogram with
perpendicular, congruent
diagonals, then the
quadrilateral is a square.
If . . .
ABCD is a ▱, AC # BD, and
AC ≅ BD
A
D
hsm11gmse_0604_t06028
Then . . .
ABCD is a square
A
D
B
C
E
B
C
For a proof of Theorem 6-19, see Problem 2.
Problem 1
TEKS Process Standard (1)(G)
Proof Proving Theorem 6-16
How can knowing
that the quadrilateral
is a parallelogram
help you prove the
theorem?
You can use any of
the properties of
parallelograms to
help you.
276
A
D
Given: ABCD is a parallelogram, AC # BD
Prove: ABCD is a rhombus.
Since ABCD is a parallelogram, AC and BD bisect each
other, so BE ≅ DE. Since AC # BD, ∠AED and ∠AEB are
congruent right angles. By the Reflexive Property of
Congruence, AE ≅ AE. So △AEB ≅ △AED by SAS.
Corresponding parts of congruent triangles are congruent,
so AB ≅ AD. Since opposite sides of a parallelogram are
congruent, AB ≅ DC ≅ BC ≅ AD. By definition, ABCD is
a rhombus.
Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares
E
B
C
hsm11gmse_0605_t06035.ai
Problem 2
Proof Proving Theorem 6-19
How can knowing the
figure is a rectangle
help you prove it is a
square?
A rectangle has four 90°
angles. If you know the
figure is a rectangle, you
only need to show all
sides are congruent to
prove it is a square.
A
Write a two-column proof to prove Theorem 6-19.
Given: ABCD is a parallelogram, AC # DB, and
AC ≅ DB
Prove: ABCD is a square.
D
E
B
Statements
C
Reasons
1) ABCD is a parallelogram, AC # DB, and
AC ≅ DB
1) Given
2) ABCD is a rectangle.
2) Theorem 6-18
3) ∠DAB, ∠ABC, ∠BCD, and ∠CDA are right angles.
3) Def. of a rectangle
4) ABCD is a rhombus.
4) Theorem 6-16
5) AB ≅ BC ≅ CD ≅ DA
5) Def. of a rhombus
6) ABCD is a square.
6) Def. of a square
Problem 3
TEKS Process Standard (1)(F)
Identifying Rhombuses, Rectangles, and Squares
How do you get
started?
Use the properties of
rhombuses, rectangles,
and squares and the
theorems you learned
to help you determine
whether each figure is a
rhombus, a rectangle, or
a square.
Can you conclude that quadrilateral ABCD is a rhombus, a rectangle, or a square?
If so, write a paragraph proof. If not, explain.
Given: Quadrilateral ABCD with
A Given: Quadrilateral ABCD with
B AE ≅ BE ≅ CE ≅ DE
AE ≅ CE, BE ≅ DE
Prove: ABCD is a rhombus, a rectangle, or a
square.
A
D
Prove: ABCD is a rhombus,
a rectangle, or a square.
A
E
B
D
E
C
Yes. Proof: It is given that
AE ≅ BE ≅ CE ≅ DE in quadrilateral ABCD.
By the definition of segment bisector, AC
and DB bisect each other. By Theorem 6-11,
ABCD is a parallelogram. By the definition of
congruent segments, AE = BE = CE = DE. By
the Segment Addition Postulate, AC = DB.
So AC ≅ DB by the definition of congruent
segments. Therefore, by Theorem 6-18, ABCD
is a rectangle.
B
C
No. The diagonals bisect each
other, so by Theorem 6-11,
quadrilateral ABCD is a
parallelogram. The diagonals
are not perpendicular, so
ABCD is not a rhombus or a
square. The diagonals are not
congruent, so ABCD is not a
rectangle or a square.
PearsonTEXAS.com
277
Problem 4
Using Properties of Special Parallelograms
A
Algebra For what value of x is ▱ABCD a rhombus?
D
(6x 2)
For ▱ABCD to be a
rhombus, its diagonals
must bisect a pair of
opposite angles.
Set the expressions for
m∠ABD and m∠CBD
equal to each other.
Solve for x.
m∠ABD = m∠CBD
6x - 2 = 4x + 8
B
(4x 8)
hsm11gmse_0605_t06040.ai
2x - 2 = 8
2x = 10
x = 5
Problem 5
Using Properties of Parallelograms
Community Service Builders use properties of diagonals
to “square up” rectangular shapes like building frames
and playing-field boundaries. Suppose you are on the
volunteer building team at the right. You are helping to lay
out a rectangular patio for a youth center.
How can you use properties of diagonals to locate the
A four corners?
You can use two theorems.
• Theorem 6-11: If the diagonals of a quadrilateral bisect each other, then the
quadrilateral is a parallelogram.
If a quadrilateral is
both a rectangle and
a rhombus, why is it a
square?
If a quadrilateral is
a rectangle, then its
diagonals are congruent
bisectors. If it is a rhombus,
then its diagonals are
perpendicular bisectors.
So, by Theorem 6-19, the
quadrilateral is a square.
278
• Theorem 6-18: If a quadrilateral is a parallelogram with congruent diagonals,
then the quadrilateral is a rectangle.
Step 1Cut two pieces of rope that will be the diagonals of the foundation
rectangle. Cut them the same length because of Theorem 6-18.
Step 2 Join the two pieces of rope at their midpoints because of Theorem 6-11.
Step 3Pull the ropes straight and taut. The ends of the ropes will be the corners
of a rectangle.
B Can you adapt this method slightly to stake off a square play area? Explain.
Yes, you can if you make the diagonals perpendicular. The result will be a
rectangle and a rhombus, so the play area will be square.
Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares
C
HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
For what value of x is the figure the given special parallelogram?
1.
rhombus
For additional support when
completing your homework,
go to PearsonTEXAS.com.
2. rectangle
3. rectangle
L
O
(6x 9)
(2x 39)
4.
rectangle
8x 3
4
4
4x
LN 4x 7
MO 2x 13
7
N
M
6. rectangle
5. rhombus
(5x 2)
hsm11gmse_0605_t05964.ai
3x
(4x 12)
hsm11gmse_0605_t05965.ai
(3x 6)
(8x 7)
(3x 4)
hsm11gmse_0605_t05966.ai
Analyze Mathematical Relationships (1)(F) Decide whether
7.
the given information is sufficient to show the
quadrilateral
is a rectangle. Explain.
hsm11gmse_0605_t05967.ai
a.AE ≅ CE and DE ≅ BE
hsm11gmse_0605_t05968.ai
b.AD ≅ BC, AB ≅ DC, and m∠DAB = 90
D
hsm11gmse_0605_t05969
C
E
A
c.AB } CD, AD } BC, and AC ≅ DB
d.AE ≅ CE ≅ DE ≅ BE
B
Apply Mathematics (1)(A) STEM8.
You can use a simple
device called a turnbuckle to “square up” structures
that are parallelograms. For the gate pictured at the
right, you tighten or loosen the turnbuckle on the
diagonal cable so that the rectangular frame will keep
the shape of a parallelogram when it sags. What are
two ways you can make sure that the turnbuckle
works? Explain.
9.
Explain Mathematical Ideas (1)(G) Suppose the
diagonals of a parallelogram are both perpendicular
and congruent. What type of special quadrilateral is it? Explain
your reasoning.
Can you conclude that the parallelogram is a rhombus, a rectangle, or a square?
Explain.
10.
11. hsm11gmse_0605_t05961.ai
hsm11gmse_0605_t05962.ai
12.
PearsonTEXAS.com
hsm11gmse_0605_t05963.ai
279
Create Representations to Communicate Mathematical Ideas (1)(E) Given two
segments with lengths a and b (a ≠ b), what special parallelograms meet the
given conditions? Show each sketch.
13.Both diagonals have length a.
14.The two diagonals have lengths a and b.
15.One diagonal has length a, and one side of the quadrilateral has length b.
16.Prove Theorem 6-17.
Proof Given: ABCD is a parallelogram.
A
3
AC bisects ∠BAD and ∠BCD.
Prove: ABCD is a rhombus.
A
17.Prove Theorem 6-18.
Proof
Given: ▱ABCD, AC ≅ BD
B
D
Prove: ABCD is a rectangle.
D
4
2 1
C
B
hsm11gmse_0605_t05970
C
Explain Mathematical Ideas (1)(G) Explain how to construct each figure given
its diagonals.
18.parallelogram
19.rectangle
20.rhombus
hsm11gmse_0605_t05971
Determine whether the quadrilateral can be a parallelogram. Explain.
21.The diagonals are congruent, but the quadrilateral has no right angles.
22.Each diagonal is 3 cm long, and two opposite sides are 2 cm long.
23.Two opposite angles are right angles, but the quadrilateral is not a rectangle.
24.Justify Mathematical Arguments (1)(G) In Theorem 6-17, replace “a pair
Proof of opposite angles” with “one angle.” Write a paragraph that proves this new
statement to be true, or give a counterexample to prove it to be false.
TEXAS Test Practice
25.Each diagonal of a quadrilateral bisects a pair of opposite angles of the
quadrilateral. What is the most precise name for the quadrilateral?
A.parallelogram
B.rhombus
C.rectangle
D.not enough information
26.Given a triangle with side lengths 7 and 11, which value could NOT be the length
of the third side of the triangle?
F.
13
G.7
H.5
J.2
27.What is the sum of the measures of the exterior angles, one at each vertex, in a pentagon?
A.180
B.360
C.540
D.108
28.The midpoint of PQ is ( -1, 4). One endpoint is P( -7, 10). What are the coordinates
of endpoint Q? Explain your work.
280
Lesson 6-5 Conditions for Rhombuses, Rectangles, and Squares
6-6 Trapezoids and Kites
TEKS FOCUS
VOCABULARY
TEKS (5)(A) Investigate patterns to make
conjectures about geometric relationships,
including angles formed by parallel lines
cut by a transversal, criteria required for
triangle congruence, special segments of
triangles, diagonals of quadrilaterals, interior
and exterior angles of polygons, and special
segments and angles of circles choosing from
a variety of tools.
TEKS (1)(F) Analyze mathematical
relationships to connect and communicate
mathematical ideas.
Additional TEKS (1)(C)
•Base angles of a trapezoid – The
•Legs of a trapezoid – The legs of
base angles of a trapezoid are
the two angles that share a base
of the trapezoid.
a trapezoid are the nonparallel
sides of the trapezoid.
•Midsegment of a trapezoid –
•Bases of a trapezoid – The bases
The midsegment of a trapezoid
is the segment that joins the
midpoints of its legs.
of a trapezoid are the parallel
sides of the trapezoid.
•Isosceles trapezoid – An isosceles •Trapezoid – A trapezoid is a
trapezoid is a trapezoid with
legs that are congruent.
quadrilateral with exactly one
pair of parallel sides.
•Kite – A kite is a quadrilateral
•Analyze – closely examine
with two pairs of consecutive
sides congruent and no opposite
sides congruent.
objects, ideas, or relationships to
learn more about their nature
ESSENTIAL UNDERSTANDING
The angles, sides, and diagonals of a trapezoid have certain properties.
The angles, sides, and diagonals of a kite have certain properties.
Key Concept Trapezoids and Their Parts
Term Description
A trapezoid is a quadrilateral with exactly one pair of parallel
sides. The parallel sides of a trapezoid are called bases. The
nonparallel sides are called legs. The two angles that share a
base of a trapezoid are called base angles. A trapezoid has two
pairs of base angles.
Diagram
base
leg
leg
base angles
base angles
base
An isosceles trapezoid is a trapezoid with legs that are
congruent.
B
C
hsm11gmse_0606_t06314
A
A midsegment of a trapezoid is the segment that joins the
midpoints of its legs.
D
R
M
A
N
hsm11gmse_0606_t06315
T
P
PearsonTEXAS.com
hsm11gmse_0606_t06325
281
Theorem 6-20
Theorem
If a quadrilateral is an isosceles
trapezoid, then each pair of
base angles is congruent.
If . . .
TRAP is an isosceles trapezoid
with bases RA and TP
Then . . .
∠T ≅ ∠P, ∠R ≅ ∠A
A
R
A
R
T
T
P
P
You will prove Theorem 6-20 in Exercise 1.
hsm11gmse_0606_t06317
hsm11gmse_0606_t06317
Theorem 6-21
Theorem
If a quadrilateral is an isosceles
trapezoid, then its diagonals are
congruent.
If . . .
ABCD is an isosceles trapezoid
B
Then . . .
AC ≅ BD
B
C
A
C
A
D
D
You will prove Theorem 6-21 in Exercise 16.
hsm11gmse_0606_t06321
hsm11gmse_0606_t06324
If . . .
TRAP is a trapezoid with
midsegment MN
Then . . .
(1) MN } TP, MN } RA, and
Theorem 6-22 Trapezoid Midsegment Theorem
Theorem
If a quadrilateral is a trapezoid,
then
(1) the midsegment is parallel
to the bases, and
(2) the length of the
midsegment is half the sum of
the lengths of the bases.
R
M
T
A
(
(2) MN = 12 TP + RA
)
N
P
You will prove Theorem 6-22 in Lesson 7-3.
hsm11gmse_0606_t06325
Key Concept Kites
Term Description
A kite is a quadrilateral with two pairs
of consecutive sides congruent and no
opposite sides congruent.
282
Lesson 6-6 Trapezoids and Kites
Diagram
hsm11gmse_0606_t06333
Theorem 6-23
Theorem
If a quadrilateral is a kite, then
its diagonals are perpendicular.
If . . .
ABCD is a kite
Then . . .
AC # BD
B
B
A
A
C
D
C
D
For a proof of Theorem 6-23, see the Reference section on page 683.
hsm11gmse_0606_t06335
hsm11gmse_0606_t06336
Concept Summary Relationships Among
Quadrilaterals
s of
pair
o
es
N
l sid
e
l
l
a
par
Kite
Quadrilateral
Only 1 pair of
parallel sides
2 pair
s of
l sides
paralle
Parallelogram
Trapezoid
Rhombus
Rectangle
Isosceles
trapezoid
hsm11gmse_0606_t06342.ai
Problem
1
Square
TEKS Process Standard (1)(F)
Finding Angle Measures in Trapezoids
What do you know
about the angles
of an isosceles
trapezoid?
You know that each
pair of base angles is
congruent. Because the
bases of a trapezoid
are parallel, you also
know that two angles
that share a leg are
supplementary.
D
CDEF is an isosceles trapezoid and mjC = 65. What are mjD,
mjE, and mjF ?
m∠C + m∠D = 180
Two angles that form same-side interior
angles along one leg are supplementary.
65 + m∠D = 180
m∠D = 115
E
65
C
F
Substitute.
Subtract 65 from each side.
Since each pair of base angles of an isosceles trapezoid is congruent, m∠C = m∠F = 65 and
hsm11gmse_0606_t06318
m∠D = m∠E = 115.
PearsonTEXAS.com
283
Problem 2
Finding Angle Measures in Isosceles Trapezoids
What do you notice
about the diagram?
Each trapezoid is part of
an isosceles triangle with
base angles that are the
acute base angles of the
trapezoid.
Paper Fans The second ring of the paper fan shown
at the right consists of 20 congruent isosceles trapezoids
that appear to form circles. What are the measures of the
base angles of these trapezoids?
Step 1Find the measure of each angle at the center
of the fan. This is the measure of the vertex angle
of an isosceles triangle.
360
m∠1 = 20 = 18
Step 2Find the measure of each acute base angle of
an isosceles triangle.
18 + x + x = 180
18 + 2x = 180
2x = 162
x = 81 Triangle Angle-Sum Theorem
Combine like terms.
Subtract 18 from each side.
Divide each side by 2.
Step 3Find the measure of each obtuse base angle of
the isosceles trapezoid.
81 + y = 180
y = 99 Two angles that form same-side interior
angles along one leg are supplementary.
Subtract 81 from each side.
Each acute base angle measures 81. Each obtuse base angle measures 99.
Problem 3
TEKS Process Standard (1)(C)
Investigating the Diagonals of Isosceles Trapezoids
How is an isosceles
trapezoid different
from other
trapezoids?
An isosceles trapezoid
is a trapezoid whose
nonparallel legs are
congruent.
Choose from a variety of tools (such as a protractor, a ruler, or a compass) to
A investigate patterns in the diagonals of the three given isosceles trapezoids.
Explain your choice.
A ruler is useful for measuring segments.
B Make a conjecture about the diagonals of isosceles trapezoids.
Isosceles Trapezoid ABCD
A
AC = 3 cm and BD = 3 cm.
B
So AC = BD and AC ≅ BD.
D
C
continued on next page ▶
284
Lesson 6-6 Trapezoids and Kites
Problem 3
continued
Isosceles Trapezoid EFGH
E
Isosceles Trapezoid JKLM
F
J
EG = 2.5 cm and FH = 2.5 cm.
JL = 2 cm and KM = 2 cm.
M
So EG = FH and EG ≅ FH.So JL = KM and JL ≅ KM.
K
L
G
H
Conjecture: If a quadrilateral is an isosceles trapezoid, then its diagonals
are congruent.
Problem 4
Using the Midsegment of a Trapezoid
Algebra QR is the midsegment of trapezoid LMNP.
What is x?
QR = 12 (LM + PN)
How can you check
your answer?
Find LM and QR. Then
see if QR equals half
of the sum of the base
lengths.
4x 10
Trapezoid Midsegment
Theorem
x + 2 = 12 [(4x - 10) + 8]
Substitute.
x + 2 = 12 (4x - 2)
Simplify.
x + 2 = 2x - 1
Distributive Property
3 = x
M
L
Q
x2
P
R
N
8
Subtract x and add 1 to each side.
Problem 5
hsm11gmse_0606_t06328
Finding Angle Measures in Kites
D
Quadrilateral DEFG is a kite. What are mj1, mj2, and mj3?
How are the triangles
congruent by SSS?
DE ≅ DG and FE ≅ FG
because a kite has
congruent consecutive
sides. DF ≅ DF by the
Reflexive Property of
Congruence.
m∠1 = 90 90 + m∠2 + 52 = 180
142 + m∠2 = 180
m∠2 = 38 Diagonals of a kite are #.
3
E
52
1 2
G
Triangle Angle-Sum Theorem
Simplify.
Subtract 142 from each side.
F
△DEF ≅ △DGF by SSS. Since corresponding parts of congruent
triangles are congruent, m∠3 = m∠GDF = 52.
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HO
ME
RK
O
NLINE
WO
PRACTICE and APPLICATION EXERCISES
Scan page for a Virtual Nerd™ tutorial video.
1.Justify Mathematical Arguments (1)(G) The plan suggests a proof of
Proof Theorem 6-20. Write a proof that follows the plan.
For additional support when
completing your homework,
go to PearsonTEXAS.com.
A
Given: Isosceles trapezoid ABCD with AB ≅ DC
Prove: ∠B ≅ ∠C and ∠BAD ≅ ∠D
D
1
B
E
Plan: Begin by drawing AE } DC to form parallelogram AECD so that AE ≅ DC ≅ AB.
∠B ≅ ∠C because ∠B ≅ ∠1 and ∠1 ≅ ∠C. Also, ∠BAD ≅ ∠D because they
are supplements of the congruent angles, ∠B and ∠C.
C
Analyze Mathematical Relationships (1)(F) Find the value(s) of the variable(s)hsm11gmse_0606_t06008
in each
isosceles trapezoid or kite.
2.
3.
4.
Q
(3x 5)
R
y
P
S
QS x 5
RP 3x 3
(x 6)
2x
(2y 20)
(2x 4)
(4x 30)
5.
Explain Mathematical Ideas (1)(G) If KLMN is an isosceles trapezoid, is it possible
for KM to bisect ∠LMN and ∠LKN ? Explain.
STEM
pplyhsm11gmse_0606_t06001
A
Mathematics (1)(A) The beams
of the
hsm11gmse_0606_t06005
bridge at the right form quadrilateral ABCD.
△AED @ △CDE @ △BEC and mjDCB = 120.
6.
Classify the quadrilateral. Explain your reasoning.
hsm11gmse_0606_t06006
A
B
E
7.
Find the measures of the other interior angles of
the quadrilateral.
D
8.
The perimeter of a kite is 66 cm. The length of one
of its sides is 3 cm less than twice the length of
another. Find the length of each side of the kite.
C
9.Prove the converse of Theorem 6-20: If a trapezoid has a pair of congruent base
Proof angles, then the trapezoid is isosceles.
Name each type of special quadrilateral that can meet the given condition.
Make sketches to support your answers.
10.exactly one pair of congruent sides
11.two pairs of parallel sides
12.four right angles
13.adjacent sides that are congruent
14.perpendicular diagonals
15.congruent diagonals
B
16.Prove Theorem 6-21.
C
Proof
Given: Isosceles trapezoid ABCD with AB ≅ DC
Prove: AC ≅ DB
286
A
D
Lesson 6-6 Trapezoids and Kites
hsm11gmse_0606_t06010
17.Prove the converse of Theorem 6-21: If the diagonals of a trapezoid are congruent,
Proof then the trapezoid is isosceles.
T
P
18.Given: Isosceles trapezoid TRAP with TR ≅ PA
Proof
Prove: ∠RTA ≅ ∠APR
19.Prove that the angles formed by the noncongruent sides of a
Proof kite are congruent.
A
R
Determine whether each statement is true or false. Justify your response.
20.All squares are rectangles.
21.A trapezoid is a parallelogram.
22.A rhombus can be a kite.
23.Some parallelograms arehsm11gmse_0606_t06009
squares.
24.Every quadrilateral is a parallelogram. 25.All rhombuses are squares.
26. Select Tools to Solve Problems (1)(C) A wallpaper border
pattern consists of isosceles trapezoids, each with two
diagonals separating it into four triangles as shown.
To investigate the trapezoids, choose from the following tools: protractor, ruler,
compass, or graphing calculator. Explain your choice. Then observe any patterns.
Make a conjecture about the triangles that are formed by the diagonals.
27.Given: Isosceles trapezoid TRAP with TR ≅ PA;
Proof
BI is the perpendicular bisector of RA,
intersecting RA at B and TP at I.
T
P
Prove: BI is the perpendicular bisector of TP.
A
R
< >
28.BN is the perpendicular bisector of AC at N. Describe the set of points, D, for
which ABCD is a kite.
B
For a trapezoid, consider the segment joining the midpoints of the two
N
A
given segments. How are its length and the lengths of the
two parallel sides
hsm11gmse_0606_t15810
of the trapezoid related? Justify your answer.
29.the two nonparallel sides
C
30.the diagonals
hsm11gmse_0606_t06011
TEXAS Test Practice
31.Which statement is never true?
A.Square ABCD is a rhombus.
C.Parallelogram PQRS is a square.
B.Trapezoid GHJK is a parallelogram. D.Square WXYZ is a parallelogram.
32.A quadrilateral has four congruent sides. Which name best describes the figure?
F.
trapezoid
G.parallelogram
H.rhombus
G
D
J.kite
33.Given DE is congruent to FG and EF is congruent to GD, prove ∠E ≅ ∠G.
E
PearsonTEXAS.com
F
287
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Topic 6 Review
TOPIC VOCABULARY
• base angles of a trapezoid,
p. 281
• equilateral polygon, p. 249
• isosceles trapezoid, p. 281
• bases of a trapezoid, p. 281
• consecutive angles, p. 255
• midsegment of a trapezoid,
• rectangle, p. 269
p. 281
• regular polygon, p. 249
• kite, p. 282
• opposite angles, p. 255
• rhombus, p. 269
• legs of a trapezoid, p. 281
• opposite sides, p. 255
• square, p. 269
• parallelogram, p. 255
• trapezoid, p. 281
• equiangular polygon,
p. 249
Check Your Understanding
Choose the vocabulary term that correctly completes the sentence.
1. A parallelogram with four congruent sides is a(n) ? .
2. A polygon with all angles congruent is a(n) ? .
3. Angles of a polygon that share a side are ? .
4. A(n) ? is a quadrilateral with exactly one pair of parallel sides.
6-1 The Polygon Angle-Sum Theorems
Quick Review
Exercises
The sum of the measures of the interior angles of an n-gon
is (n - 2)180. The measure of one interior angle of a regular
Find the measure of an interior angle and an exterior
angle of each regular polygon.
n-gon is
. The sum of the measures of the exterior
n
angles of a polygon, one at each vertex, is 360.
5.hexagon
(n - 2)180
Example
Find the measure of an interior angle of a regular 20‑gon.
(n - 2)180
Measure =
n
=
Corollary to the Polygon Angle-Sum
Theorem
(20 - 2)180
20
Substitute.
18
Simplify.
#
180
20 =
= 162
The measure of an interior angle is 162.
288
Topic 6 Review
6.16-gon
7.pentagon
8.What is the sum of the exterior angles for each polygon
in Exercises 5–7?
Find the measure of the missing angle.
9.
89
x
119
83
10.
122
79
z
hsm11gmse_06cr_t06359
hsm11gmse_06cr_t06357
6-2 Properties of Parallelograms
Quick Review
Exercises
Opposite sides and opposite angles of a parallelogram
are congruent. Consecutive angles in a parallelogram are
supplementary. The diagonals of a parallelogram bisect
each other. If three (or more) parallel lines cut off congruent
segments on one transversal, then they cut off congruent
segments on every transversal.
Find the measures of the numbered angles for each
parallelogram.
11. 3
38
12.
1
2
2
1 99
79
3
Example
13. 14.
3 1
2
hsm11gmse_06cr_t06361
1
3
63
hsm11gmse_06cr_t06363
Find the measures of the numbered angles in
the parallelogram.
2
1
37 2
3
56
Find the values of x and y in ▱ABCD.
Since consecutive angles are supplementary,
m∠1 = 180 - 56, or 124. Since opposite angles are
congruent, m∠2 = 56 and m∠3 = 124.
hsm11gmse_06cr_t06360
15. AB = 2y, BC = y + 3, CD = 5x - 1, DA = 2x + 4
hsm11gmse_06cr_t06365hsm11gmse_06cr_t06366
16. AB = 2y + 1, BC = y + 1, CD = 7x - 3, DA = 3x
6-3 Proving That a Quadrilateral Is a Parallelogram
Quick Review
Exercises
A quadrilateral is a parallelogram if any one of the following
is true.
Determine whether the quadrilateral must be a
parallelogram.
17. • Both pairs of opposite sides are parallel.
18.
• Both pairs of opposite sides are congruent.
• Consecutive angles are supplementary.
• Both pairs of opposite angles are congruent.
• The diagonals bisect each other.
• One pair of opposite sides is both congruent and
parallel.
Find the values of the variables for which ABCD must
be a parallelogram.
hsm11gmse_06cr_t06372
19. Bhsm11gmse_06cr_t06370
20. B
C
4x
(3y 20)
Example
(4y 4)
Must the quadrilateral be a parallelogram?
Yes, both pairs of opposite angles are
congruent.
4x (2x 6)
A
3y
D
3
C
2
3x
3y
1
A
D
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6-4 Properties of Rhombuses, Rectangles, and Squares
Quick Review
Exercises
A rhombus is a parallelogram with four congruent sides.
Find the measures of the numbered angles in each
special parallelogram.
A rectangle is a parallelogram with four right angles.
A square is a parallelogram with four congruent sides and
four right angles.
21. The diagonals of a rhombus are perpendicular. Each
diagonal bisects a pair of opposite angles.
2
1
32
22.
12
3
56
3
The diagonals of a rectangle are congruent.
Example
Determine whether each statement is always, sometimes,
or never true.
hsm11gmse_06cr_t06381.ai
2
What are the measures of the numbered
angles in the rhombus?
1
m∠1 = 60 Each diagonal of a rhombus
bisects a pair of opposite angles.
23. hsm11gmse_06cr_t06380.ai
A rhombus is a square.
3
60
24. A square is a rectangle.
25. A rhombus is a rectangle.
m∠2 = 90 The diagonals of a rhombus are #.
26. The diagonals of a parallelogram are perpendicular.
60 + m∠2 + m∠3 = 180 Triangle Angle-Sum Thm.
27. The diagonals of a parallelogram are congruent.
60 + 90 + m∠3 = 180 Substitute.
hsm11gmse_06cr_t06379.ai
28. Opposite angles of a parallelogram are congruent.
m∠3 = 30 Simplify.
6-5 Conditions for Rhombuses, Rectangles, and Squares
Quick Review
Exercises
If a quadrilateral is a parallelogram with a diagonal
that bisects two angles of the parallelogram, then
the quadrilateral is a rhombus. If a quadrilateral is
a parallelogram with perpendicular diagonals, then
the quadrilateral is a rhombus. If a quadrilateral is
a parallelogram with congruent diagonals, then the
quadrilateral is a rectangle.
Can you conclude that the parallelogram is a rhombus,
a rectangle, or a square? Explain.
29. 30.
Example
For what value of x is the figure the given parallelogram?
Justify your answer.
Can you conclude that the parallelogram is a rhombus,
a rectangle, or a square? Explain.
31. hsm11gmse_06cr_t06383.ai
Rhombus
32. hsm11gmse_06cr_t06384.ai
Rectangle
Yes, the diagonals are perpendicular,
so the parallelogram is a rhombus.
(5x 30) (3x 6)
1
x
2
2
2
x
3
hsm11gmse_06cr_t06385.ai
hsm11gmse_06cr_t06386.ai
hsm11gmse_06cr_t06382.ai
290
Topic 6 Review
6-6 Trapezoids and Kites
Quick Review
Exercises
The parallel sides of a trapezoid are its bases, and the
nonparallel sides are its legs. Two angles that share a
base of a trapezoid are base angles of the trapezoid.
The midsegment of a trapezoid joins the midpoints
of its legs.
Find the measures of the numbered angles in each
isosceles trapezoid.
33. 34.
1 2
1
The base angles of an isosceles trapezoid are congruent.
The diagonals of an isosceles trapezoid are congruent.
80
The diagonals of a kite are perpendicular.
Find the measures of the numbered angles in each kite.
hsm11gmse_06cr_t06388.ai
36.
35. Example
B
ABCD is an isosceles trapezoid.
What is m∠C?
Since BC } AD, ∠C and ∠D are
same-side interior angles.
m∠C + m∠D = 180
m∠C + 60 = 180
m∠C = 120
C
34
1
2
1
38
hsm11gmse_06cr_t06389.ai
2
65
A
S ame-side interior angles are
supplementary.
Substitute.
2
3
3
45
60
D
37. A trapezoid has base lengths of (6x - 1) units and
3 units. Its midsegment has a length
of (5x - 3) units.
hsm11gmse_06cr_t06391.ai
What is the value of x?
hsm11gmse_06cr_t06387.ai
hsm11gmse_06cr_t06390.ai
Subtract 60 from each side.
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Topic 6 TEKS Cumulative Practice
Multiple Choice
Read each question. Then write the letter of the correct
answer on your paper.
1.Which list could represent the lengths of the sides of a
triangle?
5.FGHJ is a quadrilateral. If at least one pair of opposite
angles in quadrilateral FGHJ is congruent, which
statement is false?
A.
Quadrilateral FGHJ is a trapezoid.
B.
Quadrilateral FGHJ is a rhombus.
A.
7 cm, 10 cm, 25 cm
C.
Quadrilateral FGHJ is a kite.
B.
4 in., 6 in., 10 in.
D.
Quadrilateral FGHJ is a parallelogram.
6.For which value of x are lines g and h parallel?
C.
1 ft, 2 ft, 4 ft
D.
3 m, 5 m, 7 m
(2x 10)
(5x 5)
2.Which quadrilateral CANNOT contain four right angles?
F.
squareH.
trapezoid
h
G.
rhombusJ.
rectangle
F.
12H.
18
3.What is the circumcenter of △ABC with vertices
A( -7, 0), B( -3, 8), and C( -3, 0)?
A.
( -7, -3)C.
( -4, 3)
B.
( -5, 4)D.
( -3, 4)
4.ABCD is a rhombus. To prove that the diagonals of a
rhombus are perpendicular, which pair of angles below
must you prove congruent by using corresponding
parts of congruent triangles?
A
B
G.
15J.
25
7.In △GHJ, GH ≅ HJ . Using the indirect proof method,
hsm11gmse_06cu_t06104
you attempt
to derive a contradiction by.ai
proving that
∠G and ∠J are right angles. Which theorem will
contradict this claim?
A.
Triangle Angle-Sum Theorem
B.
Side-Angle-Side Theorem
C.
Converse of the Isosceles Triangle Theorem
D.
Angle-Angle-Side Theorem
8.Which quadrilateral must have congruent diagonals?
E
F.
kite
D
G.
rectangle
C
F.
∠AEB and ∠DEC
G.
∠AEB and ∠AED
H.
∠BEC and ∠AED
hsm11gmse_06cu_t06102.ai
J.
∠DAB and ∠ABC
292
g
Topic 6 TEKS Cumulative Practice
H.
parallelogram
J.
rhombus
9.What values of x and y make the quadrilateral below a
parallelogram?
y
1
5x 2
4
Constructed Response
15. What are the possible values for n to make ABC a valid
hsm11gmse_06cu_t06110.ai
triangle? Show your work.
x = 2, y = 1C.
x = 1, y = 2
A.
9
x = 3, y = 5D.
B.
x = 2, y = 7
C
F.
△ABC is not isosceles.
G.
△ABC is isosceles.
H.
△ABC may or may not be isosceles.
J.
△ABC is equilateral.
Gridded Response
11. What is m∠1 in the figure below?
31
38
1
69
n
If a triangle is equilateral, then it is isosceles. △ABC is
not equilateral.
2n
1
10. Which
is the most valid conclusion based
hsm11gmse_06cu_t06105
.ai on the
statements below?
x
6y 3x 6
14. The outer walls of the Pentagon
in Arlington, Virginia, are formed
by two regular pentagons, as
shown at the right. What is the
value of x?
A
B
5n 4
16.The pattern of a soccer
ball contains regular
hexagons and regular
pentagons. The figure
at the righthsm11gmse_06cu_t06112.ai
shows what
a section of the pattern
would look like on a flat
surface. Use the fact that
there are 360° in a circle
to explain why there are
gaps between the hexagons.
Does the information help you
A
B
prove that ABCD is a
parallelogram? Explain.
hsm11gmse_06cu_t06113.ai
17. AC bisects BD.
D
C
12. ∠ABE and ∠CBD are vertical angles, and both are
complementary with ∠FGH. If m∠ABE = (3x - 1), and
m∠FGH = 4x, what is m∠CBD?
18. AB ≅ DC , AB } DC
13. What is thehsm11gmse_06cu_t06108
value of x in the kite below? .ai
20. ∠DAB ≅ ∠BCD , ∠ABC ≅ ∠CDA
21. CD has endpoints C(5, 7) and D(10, -5). What are the
coordinates of the midpoint of CD? What is CD? Show
your work.
22
19. AB ≅ DC , BC ≅ AD
hsm11gmse_06ct_t05844.ai
x
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