Download 24.3 Properties of Rectangles, Rhombuses, and Squares

LESSON
24.3
Properties of
Rectangles,
Rhombuses,
and Squares
Name
Class
Date
24.3 Properties of Rectangles,
Rhombuses, and Squares
HARDBOUND SE
PAGE 987
BEGINS HERE
Essential Question: What are the properties of rectangles, rhombuses, and squares?
Resource
Locker
Explore
Common Core Math Standards
Exploring Sides, Angles, and Diagonals of a
Rectangle
A rectangle is a quadrilateral with four right angles.
The figure shows rectangle ABCD.
The student is expected to:
Investigate properties of rectangles.
G-CO.11
A
Prove theorems about parallelograms. Also G-SRT.5
Mathematical Practices
A
B
D
C
Use a tile or pattern block and the following method to draw three different rectangles on a
separate sheet of paper.
MP.6 Precision
A
B
Language Objective
Explain to a partner how to classify different types of quadrilaterals as
rectangles, rhombuses, or squares.
D
ENGAGE
A rectangle is a parallelogram with congruent
diagonals; a rhombus is a parallelogram with
perpendicular diagonals, each of which bisects a
pair of opposite angles; a square is both a rectangle
and a rhombus and has the properties of each.
© Houghton Mifflin Harcourt Publishing Company
Essential Question: What are the
properties of rectangles, rhombuses,
and squares?
B
View the Engage section online. Discuss the photo,
drawing attention to the geometrical shapes that form
the design of the flag. Then preview the Lesson
Performance Task.
Use a ruler to measure the sides and diagonals of each rectangle. Keep track of the
measurements and compare your results to other students.
Reflect
1.
Why does this method produce a rectangle? What must you assume about the tile?
You must assume that the corners of the tile are right angles. Therefore, each stage of
the drawing produces a line that meets the previous line at a right angle. The completed
quadrilateral has four right angles, so it is a rectangle.
2.
Discussion Is every rectangle also a parallelogram? Make a conjecture based upon your measurements
and explain your thinking.
Yes; in every case, opposite sides are congruent. All rectangles are parallelograms because
a quadrilateral in which both pairs of opposite sides are congruent is a parallelogram.
3.
PREVIEW: LESSON
PERFORMANCE TASK
Use your measurements to make two conjectures about the diagonals of a rectangle.
Conjecture: The diagonals of a rectangle are congruent.
Conjecture: The diagonals of a rectangle bisect each other.
Module 24
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IN1_MNLESE389762_U9M24L3 1217
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
Module 24
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IN1_MNL
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Lesson 24.3
C
1217
4/18/14
8:02 PM
10/16/14 2:57 PM
Explain 1
HARDBOUND SE
Proving Diagonals of a Rectangle are Congruent
PAGE 988
BEGINS HERE
You can use the definition of a rectangle to prove the following theorems.
Properties of Rectangles
Exploring Sides, Angles, and
Diagonals of a Rectangle
If a quadrilateral is a rectangle, then it is a parallelogram.
If a parallelogram is a rectangle, then its diagonals are congruent.
Use a rectangle to prove the Properties
of Rectangles Theorems.
Example 1
A
B
A
B
INTEGRATE TECHNOLOGY
Given: ABCD is a rectangle.
_ _
Prove: ABCD is a parallelogram; AC ≅ BD.

D
Statements
C
D
C
D
Students have the option of doing the rectangle
activity either in the book or online.
C
Reasons
1. ABCD is a rectangle.
1. Given
2. ∠A and ∠C are right angles.
2. Definition of rectangle
3. ∠A ≅ ∠C
QUESTIONING STRATEGIES
3. All right angles are congruent.
4. ∠B and ∠D are right angles.
4. Definition of rectangle
5. ∠B ≅ ∠D
5. All right angles are congruent.
Is a rectangle a parallelogram? How do you
know? Yes; since the opposite sides are
congruent, the Opposite Sides Criterion applies.
6. ABCD is a parallelogram.
6. If both pairs of opposite angles of a quadrilateral are
congruent, then the quadrilateral is a parallelogram.
― ―
7. AD ≅ CB
7. If a quadrilateral is a parallelogram, then its opposite sides are
congruent.
8. DC ≅ DC
8. Reflexive Property of Congruence
9. ∠D and ∠C are right angles.
9. Definition of rectangle
―
―
10. ∠D ≅ ∠C
11. △ADC ≅ △BCD
― ―
12. AC ≅ BD
Proving Diagonals of a Rectangle
Are Congruent
11. SAS Triangle Congruence Theorem
12. CPCTC
A
B
Find each measure.
6.
AD = 7.5 cm and DC = 10 cm. Find DB.
D
By the Pythagorean Theorem, AC 2 = 7.5 2 + 10 2, so AC = 12.5 cm.
Diagonals of a rectangle are congruent, so DB = AC = 12.5 cm.
AB = 17 cm and BC = 12.75 cm. Find DB.
Opposite sides of the rectangle are congruent, so DC = AB = 17 cm.
By the Pythagorean Theorem, DB 2 = 17 2 + 12.75 2, so DB = 21.25 cm.
Module 24
1218
© Houghton Mifflin Harcourt Publishing Company
Discussion A student says you can also prove the diagonals are congruent in Example 1 by using the SSS
Triangle Congruence Theorem to show that △ADC ≅ △BCD. Do you agree? Explain.
No; in order to use the SSS Triangle Congruence Theorem, you would have to know that
_ _
AC ≅ BD, which is what you are trying to prove.
Your Turn
5.
EXPLAIN 1
10. All right angles are congruent.
Reflect
4.
EXPLORE
QUESTIONING STRATEGIES
What two things do you have to prove in
order to prove the theorem? Prove the figure
is a parallelogram and then prove triangles that
contain the diagonals of the rectangle are
congruent.
Does it matter which pair of triangles you
prove congruent for this proof? Explain. Yes;
you need to choose a pair of triangles that contain
the diagonals of the rectangle as sides.
C
Lesson 3
AVOID COMMON ERRORS
PROFESSIONAL DEVELOPMENT
IN1_MNLESE389762_U9M24L3.indd 1218
4/18/14 8:04 PM
Learning Progressions
In this lesson, students extend their earlier work with parallelograms to explore the
properties of three special quadrilaterals: the rectangle, a quadrilateral with four
congruent (right) angles; the rhombus, a quadrilateral with four congruent sides; and
the square, a quadrilateral with four congruent (right) angles and four congruent
sides. The proofs demonstrate that each of these quadrilaterals is a parallelogram; as
such, they inherit all of the properties of parallelograms. Proving these theorems is an
application of the Law of Detachment (if p → q is true and p is true, then q is true),
which students will use as they further their study of mathematics.
Students may try to prove that a rectangle is a
parallelogram by stating that the opposite sides are
congruent and using the Opposite Sides Criterion for
a parallelogram. However, the given information
states only that the figure is a rectangle (that is, it has
four right angles). The fact that a rectangle has
opposite sides that are congruent is a consequence of
the fact that a rectangle is a parallelogram. In order to
avoid circular reasoning, congruent opposite sides
cannot be used as part of this proof.
Properties of Rectangles, Rhombuses, and Squares
1218
HARDBOUND SE
EXPLAIN 2
Explain 2
PAGE 989
BEGINS HERE
Proving Diagonals of a Rhombus are Perpendicular
J
A rhombus is a quadrilateral with four congruent sides.
The figure shows rhombus JKLM.
Proving Diagonals of a Rhombus Are
Perpendicular
M
Properties of Rhombuses
If a quadrilateral is a rhombus, then it is a parallelogram.
If a parallelogram is a rhombus, then its diagonals are perpendicular.
If a parallelogram is a rhombus, then each diagonal bisects a pair of
opposite angles.
QUESTIONING STRATEGIES
Example 2
Why do the diagonals of a rhombus bisect
each other? Because a rhombus is a
parallelogram, the diagonals bisect each other.
Prove that the diagonals of a rhombus are perpendicular.
Given: JKLM is a rhombus.
― ―
Prove: JL ⊥ MK
M
J
N
K
L
_
To prove the diagonals of a rhombus are
perpendicular, do you need to show that each
of the four angles formed by the intersecting
diagonals is a right angle? Why or why not? No; you
need to show only that one angle is a right angle. If
one angle is a right angle, it’s easy to see that the
others must also be right angles by the Vertical
Angles Theorem and the Linear Pair Theorem.
―
― ―
Since JKLM is a rhombus, JM ≅ JK . Because JKLM is also a parallelogram, MN ≅ KN because
― ≅ JN
―,
diagonals of a parallelogram bisect each other . By the Reflexive Property of Congruence, JN
so △JNM ≅ △JNK by the SSS Triangle Congruence Theorem . So, ∠JNM ≅ ∠JNK by CPCTC.
By the Linear Pair Theorem, ∠JNM and ∠JNK are supplementary. This means that m∠JNM + m∠JNK = 180° .
Since the angles are congruent, m∠JNM = m∠JNK so by substitution , m∠JNM + m∠JNK = 180° or
_
―
2m∠JNK = 180°. Therefore, m∠JNK = 90° and JL ⊥ MK.
Reflect
© Houghton Mifflin Harcourt Publishing Company
CONNECT VOCABULARY
Have students complete a quadrilateral chart, with
pictures and explanations for each type of
quadrilateral they have encountered in this unit
(rhombus, square, rectangle, other parallelograms,
and so on).
K
L
―
7.
What can you say about the image
_of J in the proof after a reflection across
_ MK? Why?
That its image is L because MK is the perpendicular bisector of JL.
8.
What property about the diagonals of a rhombus is the same as a property
of all parallelograms? What special property do the diagonals of a rhombus have?
The diagonals of rhombuses bisect each other, which is a property of all parallelograms.
The diagonals of rhombuses are perpendicular, which is a property unique to rhombuses.
Your Turn
9.
Prove that if a parallelogram is a rhombus, then each diagonal
bisects a pair of opposite angles.
M
J
N
K
Given: JKLM is a rhombus.
―
L
Prove: MK bisects ∠JML and ∠JKL;
―
JL bisects ∠MJK and ∠MLK.
_ _
_ _ _ _
Since JKLM is a rhombus, JM ≅ LM and JK ≅ LK. MK ≅ MK by the Reflexive Property of
Congruence. So, △MJK ≅ △MLK by the SSS Triangle
_ Congruence Theorem. Therefore,
MK bisects ∠JML and ∠JKL. A similar
∠JMK ≅ ∠LMK and ∠JKM
≅
∠LKM
by
CPCTC,
so
_
argument shows that JL bisects ∠MJK and ∠MLK.
Module 24
1219
Lesson 3
COLLABORATIVE LEARNING
IN1_MNLESE389762_U9M24L3.indd 1219
Whole Class Activity
As a class, come up with a list of the properties common to some of the
quadrilaterals studied in the lesson, such as “Opposite sides congruent” and
“Diagonals perpendicular.” Use the list to construct a table summarizing which
quadrilateral (parallelogram, rectangle, rhombus, square) has each property. Save
the table as a class review resource.
1219
Lesson 24.3
4/18/14 8:04 PM
Explain 3
Example 3
Use rhombus VWXY to find each measure.
V
Y
PAGE 990
BEGINS HERE
6m - 12
(9n + 4)°
W
Z

HARDBOUND SE
Using Properties of Rhombuses to Find Measures
Using Properties of Rhombuses to
Find Measures
4m + 4
X
(3n2 - 0.75)°
Find XY.
QUESTIONING STRATEGIES
_ _
All sides of a rhombus are congruent, so VW ≅ WX and VW = WX.
What type of angles do the diagonals of a
rhombus form? right angles
6m - 12 = 4m + 4
Substitute values for VW and WX.
m=8
Solve for m.
EXPLAIN 3
VW = 6(8) - 12 = 36
_ _
Because all sides of the rhombus are congruent, then VW ≅ XY, and XY = 36.
Sustitute the value of m to find VW.

AVOID COMMON ERRORS
Find ∠YVW.
Students may be confused when asked to solve
equations to find segments of special quadrilaterals.
Remind students that the sides and angles marked in
the figure are not necessarily the ones asked for in the
problem. Have students, when they read a problem,
copy the figure and circle the length or measure they
are asked to find.
The diagonals of a rhombus are perpendicular , so ∠WZX is a right angle and
m∠WZX = 90° .
Since m∠WZX = (3n 2 - 0.75)°, then
(3n 2 - 0.75°) = 90° .
3n 2 - 0.75 = 90
Solve for n.
n = 5.5
Substitute the value of n to find m∠WVZ.
m∠WVZ = 53.5°
_
Since VX bisects ∠YVW, then ∠YVZ ≅ ∠WVZ
m∠YVW = 2(53.5°) = 107°
Substitute 53.5° for m∠WVZ.
© Houghton Mifflin Harcourt Publishing Company
Your Turn
Use the rhombus VWXY from Example 3 to find each measure.
10. Find m∠VYX.
11. Find m∠XYZ.
1 (m∠VYX)
m∠XYZ = __
2
From Part B, m∠YVW = 107°
1(
m∠XYZ = _
73°)
2
107° + m∠VYX = 180°
m∠VYX = 73°
Module 24
m∠XYZ = 36. 5°
1220
Lesson 3
DIFFERENTIATE INSTRUCTION
IN1_MNLESE389762_U9M24L3 1220
9/30/14 12:46 PM
Modeling
Divide students into groups. Have each group cut out models of a rectangle, a
rhombus, and a square from construction paper. Then have students use a
protractor and a ruler to verify the properties of rectangles and rhombuses from
the theorems in this lesson.
Properties of Rectangles, Rhombuses, and Squares
1220
Explain 4
EXPLAIN 4
Investigating the Properties of a Square
A square is a quadrilateral with four sides congruent
and four right angles.
Investigating the Properties
of a Square
Example 4

If a quadrilateral is a square, then it is a parallelogram.
By definition, a square is a quadrilateral with four congruent sides.
Any quadrilateral with both pairs of opposite sides congruent is a parallelogram,
so a square is a parallelogram.
QUESTIONING STRATEGIES
Why is a square also a rhombus? A square is a
quadrilateral with 4 congruent sides, so a
square is a rhombus.
Explain why each conditional statement is true.
HARDBOUND SE

PAGE 991
If a quadrilateral is a square, then it is a rectangle.
By definition, a square is a quadrilateral with four right angles .
BEGINS HERE
By definition, a rectangle is also a quadrilateral with four right angles .
Therefore, a square is a rectangle.
AVOID COMMON ERRORS
Your Turn
When working with shapes they are familiar with,
like squares, some students have trouble
distinguishing between properties that belong to the
shape by definition and properties that must be
proven. When discussing these definitions, point out
which properties are included and which are not.
12. Explain why this conditional statement is true: If a quadrilateral
is a square, then it is a rhombus.
By definition, a rhombus is a quadrilateral with four congruent sides. Since a square is also
a quadrilateral that has four congruent sides, then a square is a rhombus.
13. Look at Part A. Use a different way to explain why this conditional
statement is true: If a quadrilateral is a square, then it is a parallelogram.
Possible answer: All four angles of a square are right angles. All right angles are congruent.
Any quadrilateral with both pairs of opposite angles congruent is a parallelogram, so a
square is a parallelogram.
QUESTIONING STRATEGIES
How do the definitions of the special
quadrilaterals in this lesson differ from the
theorems in this lesson? The definitions only
identify the figures as quadrilaterals. Proving the
properties as theorems shows they are also
parallelograms.
© Houghton Mifflin Harcourt Publishing Company
Elaborate
ELABORATE
Quadrilateral
14. Discussion The Venn diagram shows how
quadrilaterals, parallelograms, rectangles, rhombuses,
and squares are related to each other. From this lesson,
what do you notice about the definitions and theorems
regarding these figures?
A rectangle, a rhombus, and a square are all
Parallelogram
Square
Rectangle
Rhombus
defined as quadrilaterals, but they can only be shown to be parallelograms by theorems
that prove their properties.
15. Essential Question Check-In What are the properties of rectangles
and rhombuses? How does a square relate to rectangles and rhombuses?
A rectangle is a parallelogram with four right angles and congruent diagonals. A rhombus
is a parallelogram with four congruent sides, diagonals that are perpendicular, and each of
SUMMARIZE THE LESSON
Have students make a graphic organizer or
chart to summarize properties of rectangles
and rhombuses. Sample:
its diagonals bisects a pair of opposite angles. A square is both a rectangle and a rhombus,
so it has the properties of both.
Module 24
1221
Lesson 3
LANGUAGE SUPPORT
IN1_MNLESE389762_U9M24L3.indd 1221
Rectangle
Diagonals
are congruent.
Is a
parallelogram
1221
Lesson 24.3
Rhombus
Diagonals are
perpendicular.
Diagonals bisect
a pair of opposite
angles.
Connect Vocabulary
Students may have difficulty distinguishing the special quadrilaterals in this
lesson. Have them write the definitions on a poster, and look through magazines
or books to find pictures of rectangles, rhombuses, and squares that they can add
to the poster. Have them group the pictures based on the type of figure, show the
pictures to the class, and then describe the identifying characteristics of each type
of figure.
4/18/14 8:04 PM
Evaluate: Homework and Practice
1.
EVALUATE
Complete the paragraph proof of the Properties of Rectangles Theorems.
Given: ABCD is a rectangle.
_ _
Prove: ABCD is a parallelogram; AC ≅ BD.
A
B
D
C
• Online Homework
• Hints and Help
• Extra Practice
Proof that ABCD is a parallelogram : Since ABCD is a rectangle, ∠A and
∠C are right angles. So ∠A ≅ ∠C because all right angles are congruent.
ASSIGNMENT GUIDE
By similar reasoning, ∠B ≅ ∠D. Therefore, ABCD is a parallelogram because
it is a quadrilateral that has congruent opposite angles.
Proof that the diagonals are congruent: Since ABCD is a parallelogram,
―
―
AD ≅ BC because opposite sides of a parallelogram are congruent
_ _
Also, DC ≅ DC by the Reflexive Property of Congruence. By the definition of a
.
rectangle, ∠D and ∠C are right angles, and so ∠D ≅ ∠C
because all right angles are congruent . Therefore, △ADC ≅ △BCD by the
_
SAS Triangle Congruence Theorem and _
AC ≅ BD by CPCTC.
Find the lengths using rectangle ABCD.
AB = 21; AD = 28. What is the value of AC + BD?
BD = 21 + 28 , so BD = 35. Then, AC = 35, so AC + BD = 70.
2
3.
2
2
BC = 40; CD = 30. What is the value of BD - AC?
The diagonals of the rectangle are congruent, so BD = AC.
Then, BD - AC = 0.
4.
A
D
An artist connects stained glass pieces with lead strips. In this rectangular window,
the strips are cut so that FH = 34 in. Find JG. Explain.
F
G
J
E
HARDBOUND SE
C
© Houghton Mifflin Harcourt Publishing Company • Image Credits: Courtesy of
Wimberley Stain Glass/Houghton Mifflin Harcourt Photo by Pet
2.
B
PAGE 992
BEGINS HERE
EG also measures 34 inches. Since diagonals of a parallelogram bisect each other,
1
1( )
JG = _
EG. So, JG = _
34 = 17 in.
2
2
Exercise
IN1_MNLESE389762_U9M24L3.indd 1222
Lesson 3
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–3
2 Skills/Concepts
MP.2 Reasoning
4–6
2 Skills/Concepts
MP.4 Modeling
7–19
2 Skills/Concepts
MP.2 Reasoning
20
3 Strategic Thinking
MP.3 Logic
21
3 Strategic Thinking
MP.3 Logic
Explore
Exploring Properties of Rectangles
Exercise 1
Example 1
Proving Diagonals of a Rectangle
Are Congruent
Exercises 2–6
Example 2
Proving Diagonals of a Rhombus
Are Perpendicular
Exercise 8
Example 3
Using Properties of Rhombuses to
Find Measures
Exercises 9–12
Example 4
Investigating the Properties
of a Square
Exercise 19
approach for exploring special quadrilaterals. Have
students make a physical model of a parallelogram
with paper strips and brads. Ask them to adjust the
model until it is a rectangle, as shown in an exercise,
and then measure its diagonals. Then have them
adjust the model to form a rhombus and measure the
angles made by the intersecting diagonals.
H
1222
Practice
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Some students may benefit from a hands-on
_ _
EG ≅ FH because the diagonals of a rectangle are congruent. Since FH measures 34 inches,
Module 24
Concepts and Skills
2/26/16 4:18 AM
Properties of Rectangles, Rhombuses, and Squares
1222
The rectangular gate has diagonal braces. Find each length.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 You may want to review the properties of
5.
J
H
L
30.8 in.
7.
HK = JG
= 2(JL)
= 2(30.8 in.)
= 48 in.
= 61.6 in.
Find the measure of each numbered angle in the rectangle.
∠3 and ∠5 are right angles, so they each measure 90°.
∠2 ≅ to the give angle, so m∠2 = 61°.
m∠1 = 90° - 61° = 29°; ∠4 ≅ ∠1, so m∠4 = m∠1 = 29°.
Complete the two-column proof that the diagonals
of a rhombus are perpendicular.
M
Given: JKLM is a rhombus.
― ―
Prove: JL ⊥ MK
© Houghton Mifflin Harcourt Publishing Company
_ _
1. JM ≅ JK
_ _
2. MN ≅ KN
61°
4
2
3
K
L
Statements
Reasons
1. Definition of rhombus
2. Diagonals of a parallelogram
bisect each other.
_ _
3. JN ≅ JN
3. Reflexive Property of Congruence
4. △JNM ≅ △JNK
4. SSS Triangle Congruence Theorem
5. ∠JNM ≅ ∠JNK
5. CPCTC
6. ∠JNM and ∠JNK are supplementary.
6. Linear Pair Theorem
7. m∠JNM + m∠JNK = 180°
7. Definition of supplementary
8. ∠JNM = ∠JNK
8. Definition of congruence
9. m∠JNK + ∠JNK = 180°
9. Substitution Property of Equality
10. 2m∠JNK = 180°
10. Addition
11. m∠JNK = 90°
11. Division Property of Equality
12. JL ⊥ MK
12. Definition of perpendicular lines
_
Exercise
IN1_MNLESE389762_U9M24L3 1223
J
N
ge07se_c06l04003a
AB
_
Module 24
Lesson 24.3
1
5
So, m∠1 = 29°, m∠2 = 61°, m∠3 = 90°, m∠4 = 29°, and m∠5 = 90°.
8.
1223
Find HK.
6.
HJ = KG
G 48 in. K
parallelograms before assigning the exercises. Then
summarize the properties of rectangles and
rhombuses and point out that these figures are also
parallelograms. Explain that a square has all the
properties of a parallelogram, a rectangle, and a
rhombus.
Find HJ.
_ _
HJ ≅ KG
Lesson 3
1223
Depth of Knowledge (D.O.K.)
Mathematical Practices
4/25/14 8:44 PM
22
3 Strategic Thinking
MP.3 Logic
23
3 Strategic Thinking
MP.3 Logic
ABCD is a rhombus. Find each measure.
9.
B
F
12y°
Find AB.
Possible answer:
4x + 15
PAGE 993
BEGINS HERE
7x + 2
A
4x + 15 = 7x + 2
HARDBOUND SE
C
(4y - 1)°
D
13 = 3x
1
4_
=x
3
_ _
1
1
AB ≅ CD, so AB = CD = 7x + 2 = 7 4_
+ 2 = 32_
3
3
10. Find m∠ABC.
AVOID COMMON ERRORS
Students might expect to use only the current lesson’s
properties and theorems when writing proofs. Point
out that proofs often build on previous knowledge
and thus use concepts learned in earlier lessons.v
( )
Possible answer:
_
AC ⊥ ¯
BD, so m∠AFB = 90°; 12y = 90, so y = 7.5.
m∠ABC = 180° - 2(4y -1)°
= 180° - 2(4 ⋅ 7.5 -1)° = 122°
Find the measure of each numbered angle in the rhombus.
11.
12.
1
27°
5
2 3
2
3
4
70°
4
1
5
∠2 ≅ ∠3 ≅ ∠5 ≅ to the given angle,
so they each measure 27°.
∠4 ≅ to the given angle, so m∠4 = 70°.
m∠1 + m∠2 + 27° = 180°
m∠1 + m∠2 + 70° = 180°
∠1 ≅ ∠2 ≅ ∠3 ≅ ∠5
m∠1 + 27° + 27° = 180°
m∠1 + m∠1 + 70° = 180°
∠4 ≅ ∠1, so m∠4 = 126°
So, m∠1 = 55°, m∠2 = 55°, m∠3 = 55°,
m∠4 = 70° and m∠5 = 55°.
m∠1 = 126°
m∠1 = 55°
13. Select the word that best describes when each of the following statements are true.
Select the correct answer for each lettered part.
A. A rectangle is a parallelogram.
always
sometimes
never
B. A parallelogram is a rhombus.
always
sometimes
never
C. A square is a rhombus.
always
sometimes
never
D. A rhombus is a square.
always
sometimes
never
E. A rhombus is a rectangle.
always
sometimes
never
Module 24
IN1_MNLESE389762_U9M24L3 1224
1224
© Houghton Mifflin Harcourt Publishing Company
So, m∠1 = 126°, m∠2 = 27°, m∠3 = 27°,
m∠4 = 126°, and m∠5 = 27°.
Lesson 3
9/30/14 12:49 PM
Properties of Rectangles, Rhombuses, and Squares
1224
14. Use properties of special parallelograms to complete the proof.
_
Given: EFGH is a rectangle. J is the midpoint of EH.
Prove: △FJG is isosceles.
Statements
F
E
G
H
J
Reasons
1. EFGH is a rectangle.
J is the
_
midpoint of EH.
1. Given
2. ∠E and ∠H are right angles.
2. Definition of rectangle
3. ∠E ≅ ∠H
3. All right angles are congruent.
4. EFGH is a parallelogram.
4. A rectangle is a parallelogram.
5. EF ≅ GH
5. In a parallelogram, opposite
_
_
_
_
sides are congruent.
6. EJ ≅ HJ
6. Definition of midpoint
7. △FJE ≅ △GJH
7. SAS Triangle Congruence Theorem
_ _
8. FJ ≅ GJ
9. △FJE is isosceles.
8. CPCTC
9. Definition of isosceles triangle
15. Explain the Error Find and explain the error in this paragraph proof.
Then describe a way to correct the proof.
K
© Houghton Mifflin Harcourt Publishing Company
Given: JKLM is a rhombus.
J
Prove: JKLM is a parallelogram.
Proof: It is given that JLKM is a rhombus. So, by the definition of a
rhombus,
― ―
― ―
JK ≅ LM, and KL ≅ MJ. If a quadrilateral is a parallelogram, then its opposite
sides are congruent. So JKLM is a parallelogram.
IN1_MNLESE389762_U9M24L3.indd 1225
Lesson 24.3
M
You cannot use a theorem that assumes the quadrilateral is a
parallelogram to justify the final statement because you do not know that
JKLM is a parallelogram. That is what you are trying to prove. Instead,
use the converse, which states that if both pairs of opposite sides of a
quadrilateral are congruent, then the quadrilateral is a parallelogram.
So therefore, JKLM is a parallelogram.
Module 24
1225
L
1225
Lesson 3
4/18/14 8:03 PM
HARDBOUND SE
The opening of a soccer goal is shaped like a rectangle.
PAGE 994
BEGINS HERE
16. Draw a rectangle to represent a soccer goal. Label the
rectangle ABCD
_ to show that the distance between the
goalposts, BC, is three times the distance from the top of
the goalpost to the ground. If_
the perimeter of ABCD is
64 feet, what is the length of BC?
Possible drawing:
3x
B
x
A
C
D
Perimeter = AB + BC + CD + DA
64 = x + 3x + x + 3x
64 = 8x
8=x
BC = 3x = 3(8) = 24 feet
PEERTOPEER DISCUSSION
Ask students to work with a partner to write
conditional statements in the form “If a quadrilateral
is a [figure], then it is a [figure].” An example would
be “If a quadrilateral is a rectangle, then it is a
parallelogram.” Also have them write conditional
statements in the form “If a parallelogram is a
[figure], then it is a [figure],” for example, “If a
parallelogram is a square, then it is a rhombus.”
Have them write as many of these statements as
possible and then compare their statements with
other student pairs.
17. In your rectangle from Evaluate 16, suppose the distance from B to D is (y + 10) feet,
_
and the distance from A to C is (2y - 5.3) feet. What is the approximate length of AC?
_
_
The diagonals of a rectangle are congruent, so AC ≅ BD.
AC = BD
2y - 5.3 = y + 10
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©belterz/
iStockPhoto.com
y = 15.3
So, AC = 2(15.3) - 5.3 = 30.6 - 5.3 = 25.3 feet.
(7b - 5) meters and QR_
= (2b - 0.5)
18. PQRS is a rhombus, with PQ =_
meters. If S is the midpoint of RT, what is the length of RT?
Possible answer:
Q
P
R
_ _
By the midpoint, TS ≅ RS.
_ _ _ _
By the definition of a rhombus, PQ ≅ QR ≅ RS ≅ SP.
PQ = QR
S
T
7b - 5 = 2b − 0.5
b = 0.9
RT = TS + RS = 2(RS) = 2(QR)
= 2(2b − 0.5) = 2(2(0.9) − 0.5) = 2(1.8 − 0.5) = 2(1.3) = 2.6
_
So, the length of RT is 2.6 meters.
Module 24
IN1_MNLESE389762_U9M24L3.indd 1226
1226
Lesson 3
4/18/14 8:03 PM
Properties of Rectangles, Rhombuses, and Squares
1226
19. Communicate Mathematical Ideas List the properties that a square “inherits”
because it is each of the following quadrilaterals.
JOURNAL
Have students describe the properties of a rectangle
and a rhombus, then have them draw and label
examples of each type of figure.
a. a parallelogram
• Both pairs of opposite sides are parallel.
• Both pairs of opposite sides are congruent. • Both pairs of opposite angles are congruent. • One angle is supplementary to both of its consecutive angles. • The diagonals bisect each other.
b. a rectangle
• The diagonals are congruent.
c.
a rhombus
• The diagonals are perpendicular. • Each diagonal bisects a pair of opposite angles.
H.O.T. Focus on Higher Order Thinking
Justify Reasoning For the given figure, describe any rotations or reflections that
would carry the figure onto itself. Explain.
20. A rhombus that is not a square
180° rotation around its center; reflectional symmetry across a line that contains opposite vertices (two lines)
21. A rectangle that is not a square
© Houghton Mifflin Harcourt Publishing Company
180° rotation around its center; reflectional symmetry across a line that contains the midpoints of opposite sides (two lines)
22. A square
90° rotation around its center; reflectional symmetry across a line that contains opposite vertices (two lines), or a line that contains the midpoints of opposite sides (two lines)
23. Analyze Relationships Look at your answers for Exercises 20–22. How does your
answer to Exercise 22 relate to your answers to Exercises 20 and 21? Explain.
A square has the reflectional properties of both a rhombus that is not a square and a rectangle that is not a square. Because squares have all angles congruent and all sides congruent, as opposed to only all sides congruent (rhombuses) or only all angles congruent (rectangles), a square has 90° rotational symmetry.
Module 24
IN1_MNLESE389762_U9M24L3.indd 1227
1227
Lesson 24.3
1227
Lesson 3
4/18/14 8:03 PM
Lesson Performance Task
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Patterns
MP.8 Have students look at the rhombus on the
The portion of the Arkansas state flag that is not red is a rhombus. On one flag, the
diagonals of the rhombus measure 24 inches and 36 inches. Find the area of the rhombus.
Justify your reasoning.
Arkansas flag with diagonals measuring 24 inches
and 36 inches. Pose these questions:
• Suppose you cut the rhombus along its diagonals
to make four shapes, and then rearranged the
shapes to form another type of quadrilateral that
you studied in this lesson. Name the quadrilateral
and its dimensions. a rectangle measuring 18
in. × 24 in. or 12 in. × 36 in.
432 in. 2
Possible answer: Draw the diagonals of the rhombus. Since they are
perpendicular, the four triangles into which you have divided the rhombus
1
1 ( )( )
are right triangles. The area of each is _
bh = _
18 12 = 108 in 2. There are
2
2
four such triangles with a total area of 4 ∙ 108 = 432 in 2.
12 in.
□
Module 24
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©David H.
Lewis/E+/Getty Images
18 in.
1228
• Find the area of the new quadrilateral. Then
propose a method for finding the area of a
rhombus if you know the lengths of its
diagonals. Divide the product of the diagonals
by 2.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 What is the relationship between the
combined areas of the four red triangles on an
Arkansas flag, and the area of the rest of the flag?
Answer without referring to the actual dimensions of
the flag. They are equal. Sample answer: The four
red triangles can be rearranged to form the
rhombus that makes up the rest of the flag. So, the
two are equal in area.
Lesson 3
EXTENSION ACTIVITY
IN1_MNLESE389762_U9M24L3.indd 1228
The design of the Arkansas flag is composed of a number of geometrical shapes.
Ask students to write and solve at least four problems involving the names,
perimeters, areas, and/or other information relating to the shapes on the flag.
Students should continue to use 24 inches and 36 inches as the lengths of the
diagonals of the larger rhombus.
4/18/14 8:03 PM
Scoring Rubric
2 points: Student correctly solves the problem and explains his/her reasoning.
1 point: Student shows good understanding of the problem but does not fully
solve or explain.
0 points: Student does not demonstrate understanding of the problem.
Properties of Rectangles, Rhombuses, and Squares
1228