correctionkey=nl-b;ca-b 24 . 3 do not edit--changes …€¦ · essential question: what are the...
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Explore Exploring Sides, Angles, and Diagonals of a Rectangle
A rectangle is a quadrilateral with four right angles. The figure shows rectangle ABCD.
Investigate properties of rectangles.
A Use a tile or pattern block and the following method to draw three different rectangles on a separate sheet of paper.
B 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?
2. Discussion Is every rectangle also a parallelogram? Make a conjecture based upon your measurements and explain your thinking.
3. Use your measurements to make two conjectures about the diagonals of a rectangle.
Conjecture:
Conjecture:
Resource Locker
A B
D C
A B
D C
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.
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.
The diagonals of a rectangle bisect each other.
The diagonals of a rectangle are congruent.
Module 24 1217 Lesson 3
24.3 Properties of Rectangles, Rhombuses, and Squares
Essential Question: What are the properties of rectangles, rhombuses, and squares?
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Common Core Math StandardsThe student is expected to:
G-CO.11
Prove theorems about parallelograms. Also G-SRT.5
Mathematical Practices
MP.6 Precision
Language ObjectiveExplain to a partner how to classify different types of quadrilaterals as rectangles, rhombuses, or squares.
HARDBOUND SEPAGE 987
BEGINS HERE
HARDCOVER PAGES 987994
Turn to these pages to find this lesson in the hardcover student edition.
Properties of Rectangles, Rhombuses, and Squares
ENGAGE Essential Question: What are the properties of rectangles, rhombuses, and squares?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.
PREVIEW: LESSON PERFORMANCE TASKView 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.
1217
HARDCOVER
Turn to these pages to find this lesson in the hardcover student edition.
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Explore Constructing Perpendicular Bisectors
and Perpendicular Lines
A rectangle is a quadrilateral with four right angles.
The figure shows rectangle ABCD.
Investigate properties of rectangles.
Use a tile or pattern block and the following method to draw three different rectangles on a
separate sheet of paper.
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?
2. Discussion Is every rectangle also a parallelogram? Make a conjecture based upon your measurements
and explain your thinking.
3. Use your measurements to make two conjectures about the diagonals of a rectangle.
Conjecture:
Conjecture:
Resource
Locker
G-CO.11 Prove theorems about parallelograms. Also G-SRT.5
AB
DC
A B
D C
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.
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.
The diagonals of a rectangle bisect each other.The diagonals of a rectangle are congruent.
Module 24
1217
Lesson 3
24 . 3 Properties of Rectangles,
Rhombuses, and Squares
Essential Question: What are the properties of rectangles, rhombuses, and squares?
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1217 Lesson 24 . 3
L E S S O N 24 . 3
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Explain 1 Proving Diagonals of a Rectangle are CongruentYou can use the definition of a rectangle to prove the following theorems.
Properties of Rectangles
If a quadrilateral is a rectangle, then it is a parallelogram.If a parallelogram is a rectangle, then its diagonals are congruent.
Example 1 Use a rectangle to prove the Properties of Rectangles Theorems.
Given: ABCD is a rectangle.
Prove: ABCD is a parallelogram; _ AC ≅ _ BD .
Statements Reasons
1. ABCD is a rectangle. 1. Given
2. ∠A and ∠C are right angles. 2. Definition of
3. ∠A ≅ ∠C 3. All right angles are congruent.
4. ∠B and ∠D are right angles. 4.
5. ∠B ≅ ∠D 5.
6. ABCD is a parallelogram. 6.
7. ― AD ≅
― CB 7. If a quadrilateral is a parallelogram, then its opposite sides are
congruent.
8. ― DC ≅
― DC 8.
9. ∠D and ∠C are right angles. 9. Definition of rectangle
10. ∠D ≅ ∠C 10. All right angles are congruent.
11. 11.
12. 12.
Reflect
4. 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.
Your Turn
Find each measure.
5. AD = 7.5 cm and DC = 10 cm. Find DB.
6. AB = 17 cm and BC = 12.75 cm. Find DB.
A A
D D D
B B
C CC
A
D
B
C
Reflexive Property of Congruence
△ADC ≅ △BCD
― AC ≅ ― BD
SAS Triangle Congruence Theorem
CPCTC
rectangle
Definition of rectangle
All right angles are congruent.If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.
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.
By the Pythagorean Theorem, A C 2 = 7. 5 2 + 1 0 2 , so AC = 12.5 cm. Diagonals of a rectangle are congruent, so DB = AC = 12.5 cm.
Opposite sides of the rectangle are congruent, so DC = AB = 17 cm. By the Pythagorean Theorem, D B 2 = 1 7 2 + 12.7 5 2 , so DB = 21.25 cm.
Module 24 1218 Lesson 3
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Learning ProgressionsIn 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.
HARDBOUND SEPAGE 988
BEGINS HERE EXPLORE Exploring Sides, Angles, and Diagonals of a Rectangle
INTEGRATE TECHNOLOGYStudents have the option of doing the rectangle activity either in the book or online.
QUESTIONING STRATEGIESIs a rectangle a parallelogram? How do you know? Yes; since the opposite sides are
congruent, the Opposite Sides Criterion applies.
EXPLAIN 1 Proving Diagonals of a Rectangle Are Congruent
QUESTIONING STRATEGIESWhat 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.
AVOID COMMON ERRORSStudents 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.
PROFESSIONAL DEVELOPMENT
Properties of Rectangles, Rhombuses, and Squares 1218
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Explain 2 Proving Diagonals of a Rhombus are PerpendicularA rhombus is a quadrilateral with four congruent sides. The figure shows rhombus JKLM.
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.
Example 2 Prove that the diagonals of a rhombus are perpendicular.
Given: JKLM is a rhombus.
Prove: ― JL ⊥ ― MK
Since JKLM is a rhombus, ― JM ≅ . Because JKLM is also a parallelogram, ― MN ≅ ― KN because
. By the Reflexive Property of Congruence, ― JN ≅ ― JN ,
so △JNM ≅ △JNK by the . So, by CPCTC.
By the Linear Pair Theorem, ∠JNM and ∠JNK are supplementary. This means that m∠JNM + m∠JNK = .
Since the angles are congruent, m∠JNM = m∠JNK so by , m∠JNM + m∠JNK = 180° or
2m∠JNK = 180°. Therefore, m∠JNK = and ⊥ ― MK .
Reflect
7. What can you say about the image of J in the proof after a reflection across ― MK ? Why?
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?
Your Turn
9. Prove that if a parallelogram is a rhombus, then each diagonal bisects a pair of opposite angles. Given: JKLM is a rhombus.Prove: ― MK bisects ∠JML and ∠JKL; ― JL bisects ∠MJK and ∠MLK.
M K
L
J
M K
L
JN
M K
L
JN
_ JK
diagonals of a parallelogram bisect each other
SSS Triangle Congruence Theorem
substitution
∠JNM ≅ ∠JNK
180°
90° _ JL
That its image is L because _ MK is the perpendicular bisector of
_ JL .
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.
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, ∠JMK ≅ ∠LMK and ∠JKM ≅ ∠LKM by CPCTC, so
_ MK bisects ∠JML and ∠JKL. A similar
argument shows that _ JL bisects ∠MJK and ∠MLK.
Module 24 1219 Lesson 3
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HARDBOUND SEPAGE 989
BEGINS HERE
COLLABORATIVE LEARNING
Whole Class ActivityAs 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.
EXPLAIN 2 Proving Diagonals of a Rhombus Are Perpendicular
QUESTIONING STRATEGIESWhy do the diagonals of a rhombus bisect each other? Because a rhombus is a
parallelogram, the diagonals bisect each other.
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.
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).
1219 Lesson 24 . 3
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Explain 3 Using Properties of Rhombuses to Find Measures
Example 3 Use rhombus VWXY to find each measure.
Find XY.
All sides of a rhombus are congruent, so _ VW ≅ _ WX and VW = WX.
Substitute values for VW and WX. 6m - 12 = 4m + 4
Solve for m. m = 8
Sustitute the value of m to find VW. VW = 6 (8) - 12 = 36
Because all sides of the rhombus are congruent, then _ VW ≅ _ XY , and XY = 36.
Find ∠YVW.
The diagonals of a rhombus are , so ∠WZX is a right angle and
m∠WZX = .
Since m∠WZX = (3 n 2 - 0.75) °, then .
Solve for n. 3 n 2 - 0.75 = 90
n =
Substitute the value of n to find m∠WVZ.
m∠WVZ =
Since _ VX bisects ∠YVW, then
Substitute 53.5° for m∠WVZ. m∠YVW = 2 (53.5°) = 107°
Your Turn
Use the rhombus VWXY from Example 3 to find each measure.
10. Find m∠VYX. 11. Find m∠XYZ.
Y X
W
Z
(9n + 4)°
(3n2 - 0.75)°
6m - 12
4m + 4
V
(3 n 2 - 0.75°) = 90°
∠YVZ ≅ ∠WVZ
5.5
53.5°
90°
perpendicular
From Part B, m∠YVW = 107° 107° + m∠VYX = 180° m∠VYX = 73°
m∠XYZ = 1 __ 2 (m∠VYX)
m∠XYZ = 1 _ 2 (73°) m∠XYZ = 36. 5°
Module 24 1220 Lesson 3
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HARDBOUND SEPAGE 990
BEGINS HERE
DIFFERENTIATE INSTRUCTION
ModelingDivide 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.
EXPLAIN 3 Using Properties of Rhombuses to Find Measures
QUESTIONING STRATEGIESWhat type of angles do the diagonals of a rhombus form? right angles
AVOID COMMON ERRORSStudents 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.
Properties of Rectangles, Rhombuses, and Squares 1220
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Explain 4 Investigating the Properties of a SquareA square is a quadrilateral with four sides congruent and four right angles.
Example 4 Explain why each conditional statement is true.
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.
If a quadrilateral is a square, then it is a rectangle.
By definition, a square is a quadrilateral with four .
By definition, a rectangle is also a quadrilateral with four . Therefore, a square is a rectangle.
Your Turn
12. Explain why this conditional statement is true: If a quadrilateral is a square, then it 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.
Elaborate
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?
15. Essential Question Check-In What are the properties of rectangles and rhombuses? How does a square relate to rectangles and rhombuses?
Rectangle
Parallelogram
SquareRhombus
Quadrilateral
right angles
right angles
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.
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.
A rectangle, a rhombus, and a square are all
defined as quadrilaterals, but they can only be shown to be parallelograms by theorems
that prove their properties.
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
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
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HARDBOUND SEPAGE 991
BEGINS HERE
Rectangle Rhombus
Diagonalsare congruent.
Is a parallelogram
Diagonals areperpendicular.
Diagonals bisecta pair of opposite
angles.
LANGUAGE SUPPORT
Connect VocabularyStudents 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.
EXPLAIN 4 Investigating the Properties of a Square
QUESTIONING STRATEGIESWhy is a square also a rhombus? A square is a
quadrilateral with 4 congruent sides, so a
square is a rhombus.
AVOID COMMON ERRORSWhen 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.
ELABORATE 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.
SUMMARIZE THE LESSONHave students make a graphic organizer or chart to summarize properties of rectangles
and rhombuses. Sample:
1221 Lesson 24 . 3
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• Online Homework• Hints and Help• Extra Practice
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1. Complete the paragraph proof of the Properties of Rectangles Theorems.Given: ABCD is a rectangle.Prove: ABCD is a parallelogram;
_ AC ≅ _
BD .
Proof that ABCD is a : Since ABCD is a rectangle, ∠A and
∠C are right angles. So ∠A ≅ ∠C because .
By similar reasoning, ∠B ≅ ∠D. Therefore, ABCD is a parallelogram because
Proof that the diagonals are congruent: Since ABCD is a parallelogram,
― AD ≅ ― BC because .
Also, by the Reflexive Property of Congruence. By the definition of a
rectangle, ∠D and ∠C are right angles, and so
because all right angles are . Therefore, △ADC ≅ △BCD by the
and ≅ by CPCTC.
Find the lengths using rectangle ABCD.
2. AB = 21; AD = 28. What is the value of AC + BD?
3. BC = 40; CD = 30. What is the value of BD - AC?
4. 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.
Evaluate: Homework and Practice
A B
CD
A
B C
D
E
F G
H
J
parallelogram
all right angles are congruent
it is a quadrilateral that has congruent opposite angles.
opposite sides of a parallelogram are congruent
SAS Triangle Congruence Theorem
congruent
_ DC ≅ _ DC
B D 2 = 2 1 2 + 2 8 2 , so BD = 35. Then, AC = 35, so AC + BD = 70.
_ AC
_ BD
∠D ≅ ∠C
The diagonals of the rectangle are congruent, so BD = AC. Then, BD - AC = 0.
_ EG ≅
_ FH because the diagonals of a rectangle are congruent. Since FH measures 34 inches,
EG also measures 34 inches. Since diagonals of a parallelogram bisect each other,
JG = 1 _ 2 EG. So, JG = 1 _ 2 (34) = 17 in.
Module 24 1222 Lesson 3
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IN1_MNLESE389762_U9M24L3.indd 1222 2/26/16 4:18 AMExercise 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
HARDBOUND SEPAGE 992
BEGINS HERE
EVALUATE
ASSIGNMENT GUIDE
Concepts and Skills Practice
ExploreExploring Properties of Rectangles
Exercise 1
Example 1Proving Diagonals of a Rectangle Are Congruent
Exercises 2–6
Example 2Proving Diagonals of a Rhombus Are Perpendicular
Exercise 8
Example 3Using Properties of Rhombuses to Find Measures
Exercises 9–12
Example 4Investigating the Properties of a Square
Exercise 19
INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Some students may benefit from a hands-on 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.
Properties of Rectangles, Rhombuses, and Squares 1222
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The rectangular gate has diagonal braces. Find each length.
5. Find HJ. 6. Find HK.
7. Find the measure of each numbered angle in the rectangle.
8. Complete the two-column proof that the diagonals of a rhombus are perpendicular.
Given: JKLM is a rhombus.
Prove: ― JL ⊥ ― MK
Statements Reasons
1. _
JM ≅ _
JK 1. Definition of rhombus
2. _
MN ≅ _
KN 2.
3. _
JN ≅ _
JN 3. Reflexive Property of Congruence
4. 4. SSS Triangle Congruence Theorem
5. ∠JNM ≅ ∠JNK 5.
6. ∠JNM and ∠JNK are supplementary. 6.
7. 7. Definition of supplementary
8. ∠JNM = ∠JNK 8. Definition of congruence
9. + ∠JNK = 180° 9. Substitution Property of Equality
10. 2m∠JNK = 180° 10. Addition
11. m∠JNK = 90° 11. Division Property of Equality
12. 12. Definition of perpendicular lines
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H
G
L
J
K48 in.
30.8 in.
ge07se_c06l04003aAB
5
61°
1
4 3
2
M K
L
JN
HK = JG
= 2 (JL)
= 2 (30.8 in.)
= 61.6 in.
_ HJ ≅ _ KG
HJ = KG
= 48 in.
∠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°.
So, m∠1 = 29°, m∠2 = 61°, m∠3 = 90°, m∠4 = 29°, and m∠5 = 90°.
Diagonals of a parallelogram bisect each other.
Linear Pair Theorem
CPCTC
△JNM ≅ △JNK
_ JL ⊥ _ MK
m∠JNM + m∠JNK = 180°
m∠JNK
Module 24 1223 Lesson 3
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IN1_MNLESE389762_U9M24L3 1223 4/25/14 8:44 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices
22 3 Strategic Thinking MP.3 Logic
23 3 Strategic Thinking MP.3 Logic
INTEGRATE MATHEMATICAL PRACTICESFocus on Math ConnectionsMP.1 You may want to review the properties of 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.
1223 Lesson 24 . 3
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ABCD is a rhombus. Find each measure.
9. Find AB.
10. Find m∠ABC.
Find the measure of each numbered angle in the rhombus.
11. 12.
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 neverB. A parallelogram is a rhombus. always sometimes neverC. A square is a rhombus. always sometimes neverD. A rhombus is a square. always sometimes neverE. A rhombus is a rectangle. always sometimes never
B C
D
F
A
(4y - 1)°12y°7x + 2
4x + 15
527°
1
432
570° 1
432
Possible answer:
4x + 15 = 7x + 2
13 = 3x
4 1 _ 3 = x
_ AB ≅ _ CD , so AB = CD = 7x + 2 = 7 (4 1 _ 3 ) + 2 = 32 1 _ 3
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°
∠2 ≅ ∠3 ≅ ∠5 ≅ to the given angle, so they each measure 27°.m∠1 + m∠2 + 27° = 180° m∠1 + 27° + 27° = 180° m∠1 = 126° ∠4 ≅ ∠1, so m∠4 = 126°So, m∠1 = 126°, m∠2 = 27°, m∠3 = 27°, m∠4 = 126°, and m∠5 = 27°.
∠4 ≅ to the given angle, so m∠4 = 70°.∠1 ≅ ∠2 ≅ ∠3 ≅ ∠5
m∠1 + m∠2 + 70° = 180° m∠1 + m∠1 + 70° = 180° m∠1 = 55°So, m∠1 = 55°, m∠2 = 55°, m∠3 = 55°, m∠4 = 70° and m∠5 = 55°.
Module 24 1224 Lesson 3
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HARDBOUND SEPAGE 993
BEGINS HEREAVOID COMMON ERRORSStudents 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
Properties of Rectangles, Rhombuses, and Squares 1224
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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 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.
4. EFGH is a parallelogram. 4.
5. 5.
6. 6.
7. 7.
8. 8.
9. 9.
15. Explain the Error Find and explain the error in this paragraph proof. Then describe a way to correct the proof.Given: JKLM is a rhombus.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.
F G
E HJ
K L
J M
All right angles are congruent.
A rectangle is a parallelogram.
In a parallelogram, opposite sides are congruent.
Definition of midpoint
SAS Triangle Congruence Theorem
CPCTC
Definition of isosceles triangle△FJE is isosceles.
_ FJ ≅ _ GJ
_ EJ ≅
_ HJ
△FJE ≅ △GJH
_ EF ≅ _ GH
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 Lesson 3
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1225 Lesson 24 . 3
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The opening of a soccer goal is shaped like a rectangle.
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 ?
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 ?
18. PQRS is a rhombus, with PQ = (7b - 5) meters and QR = (2b - 0.5) meters. If S is the midpoint of
_ RT , what is the length of
_ RT ?
T
S
R
Q
P
Possible drawing:
Perimeter = AB + BC + CD + DA
64 = x + 3x + x + 3x
64 = 8x
8 = x
BC = 3x = 3 (8) = 24 feet
The diagonals of a rectangle are congruent, so _ AC ≅
_ BD .
AC = BD
2y - 5.3 = y + 10
y = 15.3
So, AC = 2 (15.3) - 5.3 = 30.6 - 5.3 = 25.3 feet.
Possible answer:
By the midpoint, _ TS ≅
_ RS .
By the definition of a rhombus, _ PQ ≅ _ QR ≅
_ RS ≅
_ SP .
PQ = QR
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.
B 3x
xA
C
D
Module 24 1226 Lesson 3
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BEGINS HEREPEERTOPEER DISCUSSIONAsk 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.
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19. Communicate Mathematical Ideas List the properties that a square “inherits” because it is each of the following quadrilaterals.
a. a parallelogram
b. a rectangle
c. a rhombus
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
21. A rectangle that is not a square
22. A square
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.
• 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.
• The diagonals are congruent.
• The diagonals are perpendicular.
• Each diagonal bisects a pair of opposite angles.
180° rotation around its center; reflectional symmetry across a line that contains opposite vertices (two lines)
180° rotation around its center; reflectional symmetry across a line that contains the midpoints of opposite sides (two lines)
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)
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 1227 Lesson 3
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JOURNALHave students describe the properties of a rectangle and a rhombus, then have them draw and label examples of each type of figure.
1227 Lesson 24 . 3
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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.
Lesson Performance Task
12 in.
18 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
are right triangles. The area of each is 1 _ 2 bh = 1 _ 2 (18) (12) = 108 i n 2 . There are
four such triangles with a total area of 4 ∙ 108 = 432 i n 2 .
Module 24 1228 Lesson 3
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EXTENSION ACTIVITY
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.
INTEGRATE MATHEMATICAL PRACTICESFocus on PatternsMP.8 Have students look at the rhombus on the 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.
• 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 PRACTICESFocus on ModelingMP.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.
Scoring Rubric2 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.
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