Download 24 . 1 Properties of Parallelograms

LESSON
24.1
Name
Properties of
Parallelograms
Class
Date
24.1 Properties of Parallelograms
Essential Question: What can you conclude about the sides, angles, and diagonals
of a parallelogram?
Common Core Math Standards
Explore
The student is expected to:
Resource
Locker
Investigating Parallelograms
A quadrilateral is a polygon with four sides. A parallelogram is a quadrilateral that
has two pairs of parallel sides. You can use geometry software to investigate properties
of parallelograms.
G-CO.11
Prove theorems about parallelograms. Also G-SRT.5
Mathematical Practices
MP.3 Logic
Language Objective
Explain to a partner why pictures of quadrilaterals are or are
not parallelograms.
ENGAGE

Draw a straight line. Then plot a point that is not on the line. Construct a line
through the point that is parallel to the line. This gives you a pair of parallel lines.

Repeat Step A to construct a second pair of parallel lines that intersect those from
Step A.

The intersections of the parallel lines create a parallelogram. Plot
points at these intersections. Label the points A, B, C, and D.
Opposite sides are congruent, opposite angles are
congruent, consecutive angles are supplementary,
and diagonals bisect each other.
PREVIEW: LESSON
PERFORMANCE TASK
View the Engage section online. Discuss the photo,
asking students to identify the geometric figures
formed by a scissor lift. Then preview the Lesson
Performance Task.
© Houghton Mifflin Harcourt Publishing Company
Essential Question: What can you
conclude about the sides, angles, and
diagonals of a parallelogram?
Identify the opposite sides and opposite angles of the parallelogram.
_
_
_
_
Opposite sides: Side AB is opposite side DC. Side AD is opposite side BC.
Opposite angles: ∠A is opposite ∠C. ∠B is opposite ∠D.
Module 24
be
ges must
EDIT--Chan
DO NOT Key=NL-A;CA-A
Correction
Lesson 1
1189
gh "File info"
made throu
Date
Class
Name
perties
24.1 Pro
ograms
of Parallel
s, and
sides, angle
about the
conclude
can you
ion: What
elogram?
G-SRT.5
of a parall
lograms. Also
about paralle
theorems
diagonals
Resource
Locker
Quest
Essential
G-CO.11 Prove
HARDCOVER PAGES 965974
g Para
Investigatin
lateral that
is a quadri
ties
elogram
gate proper
sides. A parallsoftware to investi
try
n with four
l is a polygo
can use geome
A quadrilateraof parallel sides. You
has two pairs s.
logram
of paralle
Explore
IN1_MNLESE389762_U9M24L1.indd 1189
s
llelogram
a line
Construct
lines.
on the line.
of parallel
that is not
you a pair
plot a point
This gives
t line. Then
to the line.
Draw a straigh
is parallel
point that
through the

n Mifflin
Harcour t
Publishin
y
g Compan

Turn to these pages to
find this lesson in the
hardcover student
edition.
© Houghto

A to
Repeat Step
Step A.
construct
a second
el lines that
pair of parall
intersect
those from
m. Plot
a parallelogra
el lines create A, B, C, and D.
the parall
the points
ctions of
_
The interse
ctions. Label
elogram.
these interse
of the parall
side BC.
_
points at
te angles_
is opposite
and opposi
Side AD
opposite sides
side DC.
_
Identify the
opposite
Side AB is
site ∠D.
sides:
∠B is oppo
site ∠C.
Opposite
oppo
is
∠A
angles:
Opposite
Lesson 1
1189
Module 24
4L1.indd
62_U9M2
ESE3897
IN1_MNL
1189
Lesson 24.1
1189
4/18/14
7:44 PM
4/18/14 7:47 PM

Measure each angle of the parallelogram.
EXPLORE
Measure the length of each side of the parallelogram. You can do this by measuring
the distance between consecutive vertices.
Investigating Parallelograms
INTEGRATE TECHNOLOGY
Students have the option of doing the parallelogram
activity either in the book or online.

Then drag the points and lines in your construction to change the shape of the
parallelogram. As you do so, look for relationships in the measurements. Make a
conjecture about the sides and angles of a parallelogram.
QUESTIONING STRATEGIES
As you drag points, does the quadrilateral
remain a parallelogram? Yes, the lines that
form opposite sides remain parallel.
Conjecture: Opposite sides of a parallelogram are congruent. Opposite
angles of a parallelogram are congruent.
A segment that connects two nonconsecutive vertices of a polygon is a diagonal.
_
¯ and BD. Plot a point at the intersection of the diagonals and
Construct diagonals AC
label it E.

¯, BE
¯, and DE
¯, CE
¯.
Measure the length of AE

Drag the points and lines in your construction to change the shape of the
parallelogram. As you do so, look for relationships in the measurements in Step G.
Make a conjecture about the diagonals of a parallelogram.
What do you notice about consecutive angles
in the parallelogram? Why does this make
sense? Consecutive angles are supplementary. This
makes sense because opposite sides are parallel, so
consecutive angles are same-side interior angles. By
the Same-Side Interior Angles Postulate, these
angles are supplementary.
© Houghton Mifflin Harcourt Publishing Company

Conjecture: The diagonals of a parallelogram bisect each other.
Reflect
1.
Consecutive angles are the angles at consecutive vertices, such as ∠A and ∠B, or ∠A
and ∠D. Use your construction to make a conjecture about consecutive angles of a
parallelogram.
Conjecture: Consecutive angles of a parallelogram are supplementary.
Module 24
1190
Lesson 1
PROFESSIONAL DEVELOPMENT
IN1_MNLESE389762_U9M24L1.indd 1190
4/18/14 7:47 PM
Math Background
In this lesson, students extend their earlier work with triangle congruence criteria
and triangle properties to prove facts about parallelograms. A parallelogram is a
quadrilateral whose opposite sides are parallel. Like every polygon, parallelograms
are named by listing consecutive vertices. Because of this convention, the pairs of
parallel sides can be identified from the parallelogram’s name. For example,
_ _ _
_
_ _
_ _
▱JKLM has sides JK , KL, LM, and MJ with JK || LM and KL || MJ . This
relationship is easily verified by sketching a parallelogram with consecutive
vertices J, K, L, and M.
Properties of Parallelograms
1190
2.
EXPLAIN 1
Proving Opposite Sides Are
Congruent
Critique Reasoning A student claims that the perimeter of △AEB in the
construction is always equal to the perimeter of △CED. Without doing any further
measurements in your construction, explain whether or not you agree with the
student’s statement.
Agree; AE = CE, BE = DE, and BA = DC, since the diagonals of the parallelogram bisect
each other and the opposite sides are congruent. So AE + EB + BA = CE + ED + DC.
Explain 1
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 The proof that the opposite sides of a
The conjecture you made in the Explore about opposite sides of a parallelogram
can be stated as a theorem. The proof involves drawing an auxiliary line in the figure.
Theorem
If a quadrilateral is a parallelogram, then its opposite sides are congruent.
parallelogram are congruent depends on students
understanding that the opposite sides of a
quadrilateral do not share a vertex (that is, they do
not intersect). Consecutive sides of a quadrilateral do
share a vertex (that is, they intersect). You may want
to help students draw and label a quadrilateral for
reference.
Example 1
Prove that the opposite sides of a parallelogram are congruent.
Given: ABCD is a parallelogram.
_ _
_ _
Prove: AB ≅ CD and AD ≅ CB
A
B
D
C
Statements
1. ABCD is a parallelogram.
_
2. Draw DB.
_ _ _ _
3. AB∥DC, AD∥BC
QUESTIONING STRATEGIES
4. ∠ADB ≅ ∠CBD
∠ABD ≅ ∠CDB
_
_
5. DB ≅ DB
© Houghton Mifflin Harcourt Publishing Company
Why do you think the proof is based on
drawing a diagonal of the parallelogram?
Drawing the diagonal creates two triangles, which
lets you use triangle congruence criteria and the
fact that corresponding parts of congruent triangles
are congruent.
Proving Opposite Sides Are Congruent
6. △ABD ≅ △CDB
_
_ _
_
7. AB ≅ CD and AD ≅ CB
Reasons
1. Given
2. Through any two points, there is exactly
one line.
3. Definition of parallelogram
4. Alternate Interior Angles Theorem
5. Reflexive Property of Congruence
6. ASA Triangle Congruence Theorem
7. CPCTC
Reflect
3.
Explain how you can use the rotational symmetry of a parallelogram to give an
argument that supports the above theorem.
Under a 180° rotation around the center, each side is mapped to its opposite side.
Since rotations preserve distance, this shows that opposite sides are congruent.
Module 24
1191
Lesson 1
COLLABORATIVE LEARNING
IN1_MNLESE389762_U9M24L1 1191
Small Group Activity
Using geometry software, have one student construct a parallelogram. Ask a
second student to measure both the opposite angles and the opposite sides of the
parallelogram to verify the corresponding theorems in this lesson. Ask a third
student to add the diagonals to the parallelogram and use the measuring features
to verify that the diagonals of a parallelogram bisect each other. Ask a fourth
student to verify that the consecutive angles of a parallelogram are supplementary
by verifying this property. As students change the parallelogram, opposite sides
and angles remain congruent.
1191
Lesson 24.1
9/30/14 12:02 PM
Explain 2
Proving Opposite Angles Are Congruent
EXPLAIN 2
The conjecture from the Explore about opposite angles of a parallelogram can also
be proven and stated as a theorem.
Proving Opposite Angles
Are Congruent
Theorem
If a quadrilateral is a parallelogram, then its opposite angles are congruent.
Prove that the opposite angles of a parallelogram are congruent.
Example 2
A
Given: ABCD is a parallelogram.
Prove: ∠A ≅ ∠C (A similar proof shows that ∠B ≅ ∠D.)
Statements
D
C
The proof that the opposite angles of a parallelogram
are congruent depends on students understanding
the difference between opposite angles and
consecutive angles. The opposite angles of a
quadrilateral do not share a side, while the
consecutive angles of a quadrilateral do share a side.
You may want to help students draw and label a
parallelogram for reference.
Reasons
1. ABCD is a parallelogram.
_
2. Draw DB.
2. Through any two points, there is exactly
_ _ _ _
3. AB∥DC, AD∥BC
3. Definition of parallelogram
4. ∠ADB ≅ ∠CBD,
4. Alternate Interior Angles Theorem
∠ABD ≅ ∠CDB
_ _
5. DB ≅ DB
CONNECT VOCABULARY
B
1. Given
one line.
5. Reflexive Property of Congruence
6. △ABD ≅ △CDB
6. ASA Triangle Congruence Theorem
7. ∠A ≅ ∠C
7. CPCTC
QUESTIONING STRATEGIES
How are the opposite angles of a parallelogram
related? They are congruent.
Reflect
4.
Explain how the proof would change in order
to prove ∠B ≅ ∠D.
_
In the second step, draw the diagonal AC. The remaining steps are all
How is the proof of this theorem similar to the
proof that the opposite sides of a
parallelogram are congruent? They both start with
drawing a diagonal and then using a triangle
congruence criterion and the fact that
corresponding parts of congruent traingles are
congruent.
similar to those in the above proof.
© Houghton Mifflin Harcourt Publishing Company
5.
In Reflect 1, you noticed that the consecutive angles of a parallelogram are
supplementary. This can be stated as the theorem, If a quadrilateral is a parallelogram,
then its consecutive angles are supplementary.
Explain why this theorem is true.
Each pair of consecutive angles of a parallelogram are same-side interior
angles for a pair of parallel lines (the opposite sides of the parallelogram),
so the angles are supplementary by the Same-Side Interior Angles Postulate.
Explain 3
Proving Diagonals Bisect Each Other
EXPLAIN 3
The conjecture from the Explore about diagonals of a parallelogram can also be proven and
stated as a theorem. One proof is shown on the facing page.
Proving Diagonals Bisect Each Other
Theorem
If a quadrilateral is a parallelogram, then its diagonals bisect each other.
Module 24
1192
Lesson 1
DIFFERENTIATE INSTRUCTION
IN1_MNLESE389762_U9M24L1.indd 1192
Modeling
Give each group a different model parallelogram made with straws. Ask each
student in the group to use measuring tools to investigate the opposite angles and
the opposite sides of their parallelogram and then record the results in a table.
Since a parallelogram is not a rigid figure, ask students to change the angle
measures of their parallelogram, redo the measurements and record their results
in the table. After several parallelograms are explored, have the group make a
conjecture about the opposite angles and opposite sides of a parallelogram, and
then present their data and conjecture to the class.
4/18/14 7:47 PM
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 As students work on the proof in this lesson,
ask them to think about how the format of the proof
makes it easier to understand the underlying
structure of the argument. Students should recognize
that a flow proof shows how one statement connects
to the next, which may not be as apparent in a
two-column format. A two-column format, on the
other hand, may make the justification for each step
clearer than a flow proof.
Properties of Parallelograms
1192
Example 3
QUESTIONING STRATEGIES
Why do you think this theorem was
introduced after the theorems about the sides
and angles of a parallelogram? The proof of this
theorem depends upon the fact that opposite sides
of a parallelogram are congruent.
Complete the flow proof that the diagonals of a parallelogram
bisect each other.
Given: ABCD is a parallelogram.
_ _
_ _
Prove: AE ≅ CE and BE ≅ DE
A
B
E
D
C
ABCD is a
parallelogram.
Given
AB ‖ DC
AVOID COMMON ERRORS
AB ≅ DC
Definition of parallelogram
Students may be confused when asked to label the
congruent parts of the parallelogram in this proof.
Have students use one color pencil to mark the pairs
of congruent sides, a second color pencil to mark the
pairs of congruent angles on the parallelogram, and a
third and fourth color pencil to mark the congruent
segments on the diagonals of the parallelogram.
Opposite sides of a parallelogram
are congruent.
∠ABE ≅ ∠CDE
∠BAE ≅ ∠DCE
Alternate Interior
Angles Theorem
Alternate Interior
Angles Theorem
△ABE ≅ △CDE
ASA Triangle Congruence
Theorem
AE ≅ CE and
BE ≅ DE
CPCTC
Reflect
© Houghton Mifflin Harcourt Publishing Company
6.
Discussion Is it possible to prove the theorem using a different triangle congruence
theorem? Explain.
Yes; ∠AEB ≅ ∠CED because they are vertical angles. Together with
¯, you can prove the theorem using the
¯ ≅ CD
∠ABE ≅ ∠CDE and AB
AAS Triangle Congruence Theorem.
Module 24
1193
Lesson 1
LANGUAGE SUPPORT
IN1_MNLESE389762_U9M24L1 1193
Communicate Math
Provide each pair of students with pictures of different quadrilaterals, including
rectangles, squares, and other parallelograms, and rulers and protractors. The first
student chooses a picture, measures angles and side lengths, and tells the second
student why this picture is or is not a parallelogram. The second student writes
notes about the picture as the first student explains. Students switch roles and
repeat the process.
1193
Lesson 24.1
9/30/14 12:05 PM
Explain 4
Using Properties of Parallelograms
EXPLAIN 4
You can use the properties of parallelograms to find unknown lengths or angle measures in a
figure.
Example 4

ABCD is a parallelogram. Find each measure.
B
AD
Use_
the fact
_that opposite sides of a parallelogram are congruent,
so AD ≅ CB and therefore AD = CB.
5x + 19
(6y + 5)°
A
(8y − 17)°
7x = 5x + 19
Write an equation.
Using Properties of Parallelograms
C
7x
AVOID COMMON ERRORS
D
As students prepare to solve these types of problems,
they may get confused about which segments or
angles correspond. Ask them to mark the congruent
parts carefully and then think about which property
applies to the information that is marked.
x = 9.5
Solve for x.
AD = 7x = 7(9.5) = 66.5

m∠B
Use the fact that opposite angles of a parallelogram are congruent,
so ∠B ≅ ∠ D and therefore m∠B = m∠ D .
Solve for y.
QUESTIONING STRATEGIES
6y + 5 = 8y - 17
Write an equation.
(( ) )
m∠B = (6y + 5)° = 6 11 + 5
°
How are the opposite sides of a parallelogram
related? How are the diagonals of a
parallelogram related? The opposite sides are
congruent and the diagonals bisect each other.
11 = y
= 71 °
Reflect
7.
Suppose you wanted to find the measures of the other angles of parallelogram ABCD.
Explain your steps.
Possible answer (using congruent opposite angles; can also be done using
supplementary consecutive angles): Since the sum of the measures of a
© Houghton Mifflin Harcourt Publishing Company
quadrilateral is 360°, m∠A + 71° + m∠C + 71° = 360°. Since opposite angles of
a parallelogram are congruent, ∠A ≅ ∠C, so m∠A + 71° + m∠A + 71° = 360°.
Solving shows that m∠A = 109°. Therefore, m∠A = m∠C = 109°.
Module 24
IN1_MNLESE389762_U9M24L1 1194
1194
Lesson 1
9/30/14 12:06 PM
Properties of Parallelograms
1194
Your Turn
ELABORATE
PQRS is a parallelogram. Find each measure.
8.
QUESTIONING STRATEGIES
QR _ _
PS ≅ QR, so PS = QR
P
2z + 4 = 3z - 4
What do we have to know first to apply any of
these theorems? The quadrilateral is a
parallelogram.
2z + 4
8= z
9.
PR
S
QR = 3z - 4 = 3(8) - 4 = 20
x+ 9
Q
T
4x - 6
3z - 4
R
_ _
PT ≅ RT, so PT = RT
x + 9 = 4x - 6
5=x
SUMMARIZE THE LESSON
PT = x + 9 = 5 + 9 = 14
PR = 2PT, so PR = 2(14) = 28
Have students make a graphic organizer to
summarize what they know about the sides,
angles, and diagonals of a parallelogram.
Elaborate
10. What do you need to know first in order to apply any of the theorems of this lesson?
You must know that the given quadrilateral is a parallelogram.
Sample:
_
11. In parallelogram ABCD, point P lies on DC, as shown in the figure.
Explain why it must be the case that DC = 2AD. Use what you know
about base angles of an isosceles triangle.
Parallelogram
A
D
B
A x°
x°
C
D
AB ≅ DC
AD ≅ BC
Opposite angles
are congruent
∠A ≅ ∠C
∠B ≅ ∠D
The diagonals
bisect each other
If E is the point where
diagonals AC and BD
intersect, then
AE ≅ CE and
BE ≅ DE
sides of a parallelogram. So, DC = DP + PC = AD + BC = AD + AD = 2AD.
12. Essential Question Check-In JKLM is a parallelogram.
J
N
Name all of the congruent segments and angles in the
figure.
M
__ __ __ _
¯≅ LM, JM ≅ LK, JN ≅ LN, MN ≅ KN, ∠KJM ≅ ∠MLK, ∠JKL ≅ ∠LMJ
JK
IN1_MNLESE389762_U9M24L1 1195
Lesson 24.1
C
Alternate Interior Angles Theorem. Therefore, ∠DAP ≅ ∠DPA. This means
_ _
_ _
_ _
△DAP is isosceles, with AD ≅ DP. Similarly, BC ≅ PC. Also, BC ≅ AD as opposite
Module 24
1195
z°
P
B
_
∠DAP ≅ ∠BAP since AP is an angle bisector. Also, ∠DPA ≅ ∠BAP by the
© Houghton Mifflin Harcourt Publishing Company
Opposite sides
are congruent
y°
y°
1195
K
L
Lesson 1
9/30/14 12:08 PM
Evaluate: Homework and Practice
1.
2.
Sabina has tiles in the shape of a parallelogram. She
labels the angles of each tile as ∠A, ∠B, ∠C, and ∠D.
Then she arranges the tiles to make the pattern shown
here and uses the pattern to make a conjecture about
opposite angles of a parallelogram. What conjecture
does she make? How does the pattern help her make the
conjecture?
EVALUATE
• Online Homework
• Hints and Help
• Extra Practice
Pablo traced along both edges of a ruler to draw two pairs
of parallel lines, as shown. Explain the next steps he could
take in order to make a conjecture about the diagonals
of a parallelogram.
_
Possible
_ answer: He can use the ruler to draw JL
and KM, label their intersection as point N, and
use the ruler to find that JN = LN and KN = MN.
His conjecture would be that the diagonals of a
parallelogram bisect each other.
K
J
N
L
M
ASSIGNMENT GUIDE
A
B A
B A
B
D
C D
C D
C
A
B A
B A
B
D
C D
C D
C
Possible conjecture: Opposite angles of a parallelogram are congruent. In the
pattern, vertical angles ∠A and ∠C, are congruent, as are vertical angles and
∠B and ∠D.
3.
Complete the flow proof that the opposite sides of a
parallelogram are congruent. Given: ABCD is a parallelogram.
_ _
_ _
Prove: AB ≅ CD and AD ≅ CB
A
B
D
C
ABCD is a
parallelogram.
Given
Draw DB.
Definition of parallelogram
Through any two points,
there is exactly one line.
∠ADB ≅ ∠CBD,
∠ABD ≅ ∠CDB
DB ≅ DB
© Houghton Mifflin Harcourt Publishing Company
AB || DC , AD || BC
Reflex. Prop. of Cong.
Alt. Int. Angles Thm.
∆ABD ≅ ∆CDB
CPCTC
Exercise
IN1_MNLESE389762_U9M24L1.indd 1196
Lesson 1
Depth of Knowledge (D.O.K.)
Mathematical Practices
1–2
2 Skills/Concepts
MP.5 Using Tools
3–5
2 Skills/Concepts
MP.3 Logic
6–9
2 Skills/Concepts
MP.4 Modeling
10–13
2 Skills/Concepts
MP.4 Modeling
14–16
2 Skills/Concepts
MP.3 Logic
17–24
2 Skills/Concepts
MP.2 Reasoning
3 Strategic Thinking
MP.4 Modeling
25
Explore
Investigating Parallelograms
Exercises 1–2
Example 1
Proving Opposite Sides
Are Congruent
Exercise 3
Example 2
Proving Opposite Angles
Are Congruent
Exercise 4
Example 3
Proving Diagonals Bisect Each Other
Exercise 5
Example 4
Using Properties of Parallelograms
Exercises 6–13
approach for finding the properties of parallelograms.
Have students copy a larger version of the
parallelogram in an exercise onto a sheet of paper
and then cut it out. Make sure the diagonals are
drawn in. Have them use a ruler to find the segment
lengths of the diagonals and verify that the diagonals
bisect each other. Students can then cut the
parallelogram along one diagonal and place the
opposite sides on top of each other to confirm that
they are coincident, and therefore congruent.
AB ≅ CD and
AD ≅ CB
1196
Practice
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Some students may benefit from a hands-on
ASA Cong. Thm.
Module 24
Concepts and Skills
4/18/14 7:47 PM
Properties of Parallelograms
1196
4.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Modeling
MP.4 Encourage students to copy the
A
Write the proof that the opposite angles of a
parallelogram are congruent as a paragraph proof.
B
D
Given: ABCD is a parallelogram.
C
Prove: ∠A ≅ ∠C (A similar proof shows that ∠B ≅ ∠D.)
parallelograms in the exercises onto their papers and
use colored pencils to keep track of the congruent
angles and segments.
5.
_ _
Possible
answer: It is given that ABCD is a parallelogram, so AB∥DC
__
and AD∥BC by the def. of a parallelogram. By the Alt. Int. Angles Thm.,
∠ADB ≅ ∠CBD
and _
∠ABD ≅ ∠CDB. Since two points determine a line, you
__
can draw DB. DB ≅ DB by the Reflex. Prop. of Cong. You can conclude that
△ABD ≅ △CDB by the ASA Cong. Thm., and so, ∠A ≅ ∠C by CPCTC.
Write the proof that the diagonals of a parallelogram
A
B
E
bisect each other as a two-column proof.
D
Given: ABCD is a parallelogram.
_ _
_ _
Prove: AE ≅ CE and BE ≅ DE
C
Statements
Reasons
1. ABCD is a parallelogram.
1. Given
_ _
2. AB ∥ DC
2. Definition of parallelogram
3. ∠ABE ≅ ∠CDE , ∠BAE ≅ ∠DCE
_ _
4. AB ≅ DC
3. Alt. Int. Angles Thm.
4. Opposite sides of a
parallelogram are congruent.
5. △ABE ≅ △CDE
_ _
_ _
6. AE ≅ CE and BE ≅ DE
5. ASA Triangle Cong. Thm.
6. CPCTC
EFGH is a parallelogram. Find each measure.
© Houghton Mifflin Harcourt Publishing Company
6.
7.
FG
_ _
HE ≅ FG; 5z - 16 = 3z + 8; z = 12; FG = 44
EG
H
― ―
EJ ≅ GJ; 4w + 4 = 2w + 22; w = 9; EJ = 40; EG = 2EJ, 80
m∠B
AD
2w + 22
3y - 1
(9x - 5)° B
D
(10x - 19)°
Exercise
IN1_MNLESE389762_U9M24L1.indd 1197
1197
Lesson 24.1
G
y + 15
― ―
AD ≅ CB; 3y - 1 = y + 15; y = 8; AD = 23
Module 24
3z + 8
A
∠B ≅ ∠D; 9x - 5 = 10x - 19; 14 = x; m∠B = 121°
9.
F
J
5z - 16
ABCD is a parallelogram. Find each measure.
8.
4w + 4
E
Lesson 1
1197
Depth of Knowledge (D.O.K.)
C
Mathematical Practices
26
3 Strategic Thinking
MP.3 Logic
27
3 Strategic Thinking
MP.2 Reasoning
4/18/14 7:47 PM
A staircase handrail is made from congruent parallelograms.
In ▱PQRS, PQ = 17.5, ST = 18, and m∠QRS = 110°.
Find each measure. Explain.
Q
10. RS
R
T
Opp. sides of PRQS are congruent, so
RS = PQ = 17.5.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Reasoning
MP.2 When working with proofs related to
P
properties of parallelograms, encourage students to
focus on the accuracy of the hypothesis and
conclusion of each statement. In this lesson, the
hypothesis of each proof includes a statement that a
quadrilateral is a parallelogram. The conclusion of
the proof will be a property of the parallelogram.
S
11. QT
The diag. of PRQS bisect each other, so QT = ST = 18.
12. m∠PQR
Consec. angles of PRQS are supplementary, so m∠PQR = 70°.
13. m∠SPQ
Opp. angles of PRQS are congruent, so m∠SPQ = m∠QRS = 110°.
Write each proof as a two-column proof.
14. Given: GHJN and JKLM are parallelograms.
H
Prove: ∠G ≅ ∠L
J
G
K
L
N
Statements
1. GHJN and JKLM are parallelograms.
3. ∠HJN ≅ ∠KJM
4. ∠G ≅ ∠L
Reasons
1. Given
2. Opp. angles of a ▱ are congruent.
3. Vertical angles are congruent.
4. Transitive Property of Congruence
_ _
15. Given: PSTV is a parallelogram. PQ ≅ RQ
Q
Prove: ∠STV ≅ ∠R
S
P
Statements
T
V
R
Reasons
1. PSTV is a parallelogram.
1. Given
2. ∠STV ≅ ∠P
2. Opp. angles of a ▱ are congruent.
―
―
3. PQ ≅ RQ
4. △PQR is isosceles.
5. ∠P ≅ ∠R
6. ∠STV ≅ ∠R
Module 24
IN1_MNLESE389762_U9M24L1.indd 1198
© Houghton Mifflin Harcourt Publishing Company • Image Credits: ©Byjeng/
Shutterstock
2. ∠G ≅ ∠HJN, ∠KJM ≅ ∠L
M
3. Given
4. Definition of isosceles triangle
5. Isosceles Triangle Theorem
6. Transitive Property of Congruence
1198
Lesson 1
4/18/14 7:46 PM
Properties of Parallelograms
1198
16. Given: ABCD and AFGH are parallelograms.
AVOID COMMON ERRORS
B
Prove: ∠C ≅ ∠G
Advise students to pay close attention to the
markings on a diagram, especially when writing a
proof. Explain that a quadrilateral with only one set
of parallel lines is not a parallelogram.
C
F
G
A
Statements
1. ABCD and AFGH are
H
D
Reasons
1. Given
parallelograms.
2. ∠C ≅ ∠A, ∠A ≅ ∠G
2. Opposite angles of a parallelogram
are congruent.
3. ∠C ≅ ∠G
3. Trans. Prop. of Cong.
Justify Reasoning Determine whether each statement is always,
sometimes, or never true. Explain your reasoning.
_ _
17. If quadrilateral RSTU is a parallelogram, then RS ≅ ST.
Sometimes; opposite sides of a parallelogram are congruent, but
―
―
consecutive sides, such as RS and ST , may or may not be congruent.
18. If a parallelogram has a 30° angle, then it also has a 150° angle.
Always; consecutive angles of a parallelogram are supplementary, so the
angle that is a consecutive angle to the 30° angle must measure 150°.
_
_
19. If quadrilateral GHJK is a parallelogram, then GH is congruent to JK .
Always; opposite sides of a parallelogram are congruent.
20. In parallelogram ABCD, ∠A is acute and ∠C is obtuse.
© Houghton Mifflin Harcourt Publishing Company
Never; opposite angles of a parallelogram are congruent; ∠A and ∠C are
opposite angles, so they must have the same measure.
_
_
21. In parallelogram MNPQ, the diagonals MP and NQ meet at R with MR = 7 cm and
RP = 5 cm.
Never; diagonals of a parallelogram bisect each other.
Module 24
IN1_MNLESE389762_U9M24L1.indd 1199
1199
Lesson 24.1
1199
Lesson 1
4/18/14 7:46 PM
22. Communicate Mathematical Ideas Explain how you can use the rotational
symmetry of a parallelogram to give an argument that supports the fact that opposite
angles of a parallelogram are congruent.
PEERTOPEER DISCUSSION
Ask students to discuss with a partner how to create a
graphic organizer that will display all the properties
of parallelograms that they have learned in the lesson.
Then have them create the graphic organizer, making
sure they include the following properties:
Under a 180° rotation around the center of the parallelogram, each angle
is mapped to its opposite angle. Since rotations preserve angle measure,
this shows that opposite angles are congruent.
23. To repair a large truck or bus, a mechanic
might use a parallelogram lift. The figure
shows a side view of the lift. FGKL, GHJK,
and FHJL are parallelograms.
F
G
1
2
5
L
3
6
H
4
7
8
K
• Opposite sides are parallel.
J
• Opposite sides are congruent.
a. Which angles are congruent to ∠1?
Explain.
_ _
∠3, ∠6, ∠8; ∠3 ≅ ∠1 since FL ǁ GK and ∠3 and ∠1 are corresponding
angles; ∠6 ≅ ∠1 since opposite angles of a parallelogram are congruent;
∠8 ≅ ∠1 since opposite angles of a parallelogram are congruent.
• Opposite angles are congruent.
• Consecutive angles are supplementary.
• Diagonals bisect each other.
b. What is the relationship between ∠1 and each of the remaining labeled angles?
Explain.
∠1 is supplementary to ∠2, ∠4, ∠5, and ∠7; ∠1 is supplementary
to ∠2, ∠4, and ∠5 since consecutive angles of a parallelogram are
supplementary. Because opposite angles of a parallelogram are
congruent, ∠7 ≅ ∠4. So if ∠1 is supplementary to ∠4, then ∠1 is
also supplementary to ∠7.
24. Justify Reasoning ABCD is a parallelogram. Determine
A
B
whether each statement must be true. Select the correct
E
answer for each lettered part. Explain your reasoning.
D
D. ∠DAC ≅ ∠BCA
E. △AED ≅ △CEB
F. ∠DAC ≅ ∠BAC
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
Yes
No
C
a. Yes; opposite sides of a parallelogram are congruent so AB = DC and AD = BC.
The perimeter of ABCD is AB + BC + DC + AD = AB + BC + AB + BC = 2AB + 2BC.
1
DB.
b. Yes; the diag. of a parallelogram bisect each other, so DE = EB and DE = _
2
c. No; opp. sides are cong., but consecutive sides may or may not be cong.
_ _
d. Yes; AD ǁ BC , so ∠DAC ≅ ∠BCA because they are alternate interior angles.
_ _
_ _
e. Yes; the diag. of a parallelogram bisect each other, so AE ≅ CE and DE ≅ BE ; ∠AED ≅ ∠CEB
because they are vert. angles; △AED ≅ △CEB by the SAS Cong. Thm.
© Houghton Mifflin Harcourt Publishing Company
A. The perimeter of ABCD is 2AB + 2BC.
1 DB
B. DE = _
_ 2_
C. BC ≅ DC
f. No; the diag. of a parallelogram bisect each other, but they may or may not bisect
opp. angles of the parallelogram.
Module 24
IN1_MNLESE389762_U9M24L1 1200
1200
Lesson 1
9/30/14 12:16 PM
Properties of Parallelograms
1200
JOURNAL
H.O.T. Focus on Higher Order Thinking
25. Represent Real-World Problems A store sells tiles in the
shape of a parallelogram. The perimeter of each tile is 29 inches.
One side of each tile is 2.5 inches longer than another side. What
are the side lengths of the tile? Explain your steps.
Have students summarize the relationships they
discovered about parallelograms.
Possible answer: Represent the side lengths as x
and (x + 2.5). Since opposite sides of a parallelogram
are congruent, the perimeter can be written as
x + (x + 2.5) + x + (x + 2.5), or 4x + 5. Write an
equation to solve for x.
4x + 5 = 29
So, x = 6, and (x + 2.5) = 8.5.
So, the side lengths of the tile are 6 inches and 8.5 inches.
26. Critique Reasoning A student claims that there is an SSSS congruence criterion for
parallelograms. That is, if all four sides of one parallelogram are congruent to the four
sides of another parallelogram, then the parallelograms are congruent. Do you agree?
If so, explain why. If not, give a counterexample. Hint: Draw a picture.
© Houghton Mifflin Harcourt Publishing Company • Image Credits:
©Tashatuvango/iStockPhoto.com
No; two parallelograms may have four congruent sides, but their four angles
do not also have to be congruent. The figures show a counterexample.
27. Analyze Relationships The figure shows two congruent
parallelograms. How are x and y related? Write an equation that
expresses the relationship. Explain your reasoning.
x°
x°
x°
x°
IN1_MNLESE389762_U9M24L1.indd 1201
Lesson 24.1
y°
y = 2x; Possible explanation: Since opposite sides of a
parallelogram are parallel, if the common side of the
parallelograms is extended, the angles formed by the
extended side each measure x° because alternate interior
angles are congruent.
Module 24
1201
x°
1201
Lesson 1
4/18/14 7:46 PM
Lesson Performance Task
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Math Connections
MP.1 An inventor considered building a scissor lift
The principle that allows a scissor lift to raise the platform on top of it to a considerable height
can be illustrated with four freezer pop sticks attached at the corners.
A
A
B
D
B
from triangles rather than parallelograms. The
inventor believed that simpler and cheaper 3-sided
shapes would appeal to budget-minded customers.
What advice would you give to the inventor and
why? Sample answer: Triangles are rigid, meaning
their shapes can’t change. A triangular scissor lift
wouldn’t work because the triangular sections could
not be made to expand or contract.
D
C
C
Answer these questions about what happens to parallelogram ABCD when you change its shape
as in the illustration.
a. Is it still a parallelogram? Explain.
b. Is its area the same? Explain.
c. Compare the lengths of the diagonals in the two figures as you
change them.
INTEGRATE MATHEMATICAL
PRACTICES
Focus on Critical Thinking
MP.3 One angle of a parallelogram is a right angle.
d. Describe a process that might be used to raise the platform on
a scissor lift.
a. Yes. Its opposite sides are still parallel and congruent.
b. No. As you move points B and D closer to each other, the area decreases.
_
_
c. Horizontal diagonal BD decreases in length. Vertical diagonal AC increases
in length.
Describe the other angles. Explain your reasoning.
The other angles are also right angles; Sample
answer: Since the opposite angles of a
parallelogram are congruent, the angle opposite
the right angle must also be a right angle. Since
consecutive angles of a parallelogram are
supplementary, the measures of the two remaining
angles must each be 180° – 90° = 90°. So, both are
also right angles.
d. Possible answer: Apply an inward horizontal force from both sides on the
bottom parallelogram.
© Houghton Mifflin Harcourt Publishing Company
Module 24
1202
Lesson 1
EXTENSION ACTIVITY
IN1_MNLESE389762_U9M24L1 1202
Supply students with sheets of light cardboard or photo-print paper, and some
brads. Instruct students to cut out twelve 5-inch-by-0.5-inch strips and to attach
the ends with brads in such a way as to make a model of a 3-parallelogram scissor
lift. Have students demonstrate the operation of the lift, describing how the
elements of the parallelogram (angle measures, side length, shape, area) change as
the lift increases and decreases in total height.
9/30/14 12:15 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 Parallelograms
1202