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Chapter 6 245 Lesson 2
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6-2
Standards for Mathematical Content
G-CO.3.11 Prove theorems about parallelograms.
G-SRT.2.5 Use congruence ... criteria for triangles to solve problems and to prove relationships in geometric figures.
Vocabularydiagonal
PrerequisitesTheorems about parallel lines cut by a transversal
Triangle congruence criteria
Math BackgroundIn this lesson, students extend their earlier work with triangle congruence criteria and triangle properties to prove facts about parallelograms. This lesson gives students a chance to use inductive and deductive reasoning to investigate properties of the sides, angles, and diagonals of parallelograms.
Students have encountered parallelograms in earlier grades. Ask a volunteer to define parallelogram. Students may have only an informal idea of what a parallelogram is (e.g., “a slanted rectangle”), so be sure they understand that the mathematical definition of a parallelogram is a quadrilateral with two pairs of parallel sides. You may want to show students how they can make a parallelogram by drawing lines on either side of a ruler, changing the position of the ruler, and drawing another pair of lines. Ask students to explain why this method creates a parallelogram.
Investigate parallelograms.
Materials: geometry software
Questioning Strategies• As you use the software to drag points
A, B, C, and/or D, does the quadrilateral remain a parallelogram? Why? Yes; the lines that form opposite sides remain parallel.
• 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.
Teaching StrategySome students may have difficulty with terms like opposite sides or consecutive angles. Remind students that 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). Opposite angles of a quadrilateral do not share a side. Consecutive angles of a quadrilateral do share a side. You may want to help students draw and label a quadrilateral for reference.
Prove that opposite sides of a parallelogram are congruent.
Questioning Strategies• Why do you think the proof is based on drawing
the diagonal ___
DB ? Drawing the diagonal creates two triangles; then you can use triangle congruence criteria and CPCTC.
INTRODUCE
TEACH
1
2
Properties of ParallelogramsFocus on ReasoningEssential question: What can you conclude about the sides, angles, and diagonals of a parallelogram?
A B
D C
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You may have discovered the following theorem about parallelograms.
Prove that opposite sides of a parallelogram are congruent.
Complete the proof.
Given: ABCD is a parallelogram.
Prove: ___
AB ≅ ____
CD and ___
AD ≅ ___
BC
REFLECT
2a. Explain how you can use the rotational symmetry of a parallelogram to give an argument that supports the above theorem.
2b. One side of a parallelogram is twice as long as another side. The perimeter of the parallelogram is 24 inches. Is it possible to find all the side lengths of the parallelogram? If so, find the lengths. If not, explain why not.
2
Statements Reasons
1. ABCD is a parallelogram. 1.
2. Draw ___
DB. 2. Through any two points there exists exactly one line.
3. ___
ABǁ ___
DC;___
ADǁ___
BC 3.
4. ∠ADB≅∠CBD;∠ABD≅∠CDB 4.
5. ___
DB≅___
DB 5.
6. 6. ASA Congruence Criterion
7. AB≅CD;AD≅BC 7.
Theorem
If a quadrilateral is a parallelogram, then opposite sides are congruent.
△ABD ≅ △CDB
Under a 180° rotation about the center of the parallelogram, each side is mapped
to its opposite side. Since rotations preserve distance, this shows that opposite
sides are congruent.
Yes; consecutive sides have lengths x, 2x, x, and 2x, so x + 2x + x + 2x = 24, or
6x = 24. Therefore x = 4 and the side lengths are 4 in., 8 in., 4 in., and 8 in.
Given
Definition of parallelogram
Alternate Interior Angles Theorem
Reflexive Property of Congruence
CPCTC
Chapter 6 246 Lesson 2
G_MFLESE200852_C06L02.indd 246 08/03/13 10:09 PM
A
AB = 2.60 cm
BC = 1.74 cmCD = 2.60 cmDA = 1.74 cm
m∠DAB = 116.72˚m∠ABC = 63.28˚m∠BCD = 116.72˚m∠CDA = 63.28˚
B
DC
A B
D C
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Name Class Date 6-2
Investigate parallelograms.
Use the straightedge tool of your geometry software to draw a straight line. Then plot a point that is not on the line. Select the point and line, go to the Construct menu, and construct a line through the point that is parallel to the line. This will give you a pair of parallel lines, as shown.
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.
Use the Measure menu to measure each angle of the parallelogram.
Use the Measure menu to measure the length of each side of the parallelogram. (You can do this by measuring the distance between consecutive vertices.)
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.
REFLECT
1a. Make a conjecture about the sides and angles of a parallelogram.
1A
B
C
D
E
F
Properties of ParallelogramsFocus on ReasoningEssential question: What can you conclude about the sides, angles, and diagonalsof a parallelogram?
Recall that a parallelogram is a quadrilateral that has two pairs of parallel sides. You use the symbol � to name a parallelogram. For example, the figure shows �ABCD.
G-CO.3.11,G-SRT.2.5
Opposite sides of a parallelogram are congruent.
Opposite angles of a parallelogram are congruent.
Chapter 6 245 Lesson 2
G_MFLBESE200852_C06L02.indd 245 03/05/14 4:56 PM
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Chapter 6 246 Lesson 2
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Notes
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Chapter 6 247 Lesson 2
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Investigate diagonals of parallelograms.
Materials: geometry software
Questioning Strategies• How many diagonals does a parallelogram have?
Is this true for every quadrilateral? Two; yes
• If a quadrilateral is named PQRS, what are the
diagonals? ___
PR and ___
QS
• Are the diagonals of a parallelogram ever
congruent? If so, when does this appear to happen?
Yes; when the parallelogram is a rectangle
Prove diagonals of a parallelogram bisect each other.
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.
TEACH
3
4
Highlighting the Standards
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. This
addresses elements of Standard 3 (Construct
viable arguments and critique the reasoning
of others). Students should recognize that a
flow proof shows how one statement connects
to the next. This may not be as apparent in a
two-column format. You may want to have
students rewrite the proof in a two-column
format as a way of exploring this further.
EA B
CD
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You may have discovered the following theorem about parallelograms.
Prove diagonals of a parallelogram bisect each other.
Complete the proof.
Given: ABCD is a parallelogram.
Prove: ___
AE � ___
CE and ___
BE � ___
DE .
REFLECT
4a. Explain how you can prove the theorem using a different congruence criterion.
C
4
Given
Definition of
parallelogram
Opposite sides of a
parallelogram are congruent.
Alternate Interior
Angles Theorem
Alternate Interior
Angles Theorem
ASA Congruence Criterion
CPCTC
Theorem
If a quadrilateral is a parallelogram, then the diagonals bisect each other.
T
∠AEB � ∠CED because they are vertical angles. Using this fact plus the fact
that ∠ABE � ∠CDE and ___
AB � ___
DC , it is possible to prove the theorem using
the AAS Congruence Criterion.
ABCD is a parallelogram.
___
AE � ___
CE and ___
BE � ___
DE .
∠ABE � ∠CDE ∠BAE � ∠DCE
�ABE � �CDE
___
AB � ___
DC ___
AB ‖ ___
DC
Chapter 6 248 Lesson 2
A B
CD
BA
CD
B
AE = 1.52 cmBE = 2.60 cmCE = 1.52 cmDE = 2.60 cm
A
C
E
D
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Investigate diagonals of parallelograms.
Use geometry software to construct a
parallelogram. (See Lesson 4-2 for detailed
instructions.) Label the vertices of the
parallelogram A, B, C, and D.
Use the segment tool to construct the
diagonals, ___
AC and ___
BD .
Plot a point at the intersection of the
diagonals. Label this point E.
Use the Measure menu to measure the
length of ___
AE , ___
BE , ___
CE , and ___
DE . (You can
do this by measuring the distance between
the relevant endpoints.)
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.
REFLECT
3a. Make a conjecture about the diagonals of a parallelogram.
3b. A student claims that the perimeter of �AEB 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.
I3A
B
C
D
E
Essential question: What can you conclude about the diagonals of a parallelogram?
A segment that connects any two nonconsecutive
vertices of a polygon is a diagonal. A
parallelogram has two diagonals. In the figure,
___
AC and ___
BD are diagonals of �ABCD.
Agree; AE = CE, BE = DE, and AB = DC since opposite sides of a parallelogram
are congruent. So, AE + BE + AB = CE + DE + DC.
The diagonals of a parallelogram bisect each other.
Chapter 6 247 Lesson 2
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Chapter 6 248 Lesson 2
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Notes
Parallelogram
A B
CD
∠A � ∠C
∠B � ∠D
Opposite angles are congruent.
AB � DC
AD � BC
Opposite sides are congruent.
AC and BD intersect,then AE � CE and
The diagonalsbisect each other.
If E is the pointwhere diagonals
BE � DE .
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Chapter 6 249 Lesson 2
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Teaching StrategyThe lesson concludes with the theorem that
states that opposite angles of a parallelogram are
congruent. The proof of this theorem is left as an
exercise (Exercise 1). Be sure students recognize
that the proof of this theorem is similar to the
proof that opposite sides of a parallelogram
are congruent. Noticing such similarities is an
important problem-solving skill.
Essential QuestionWhat can you conclude about the sides, angles, and diagonals of a parallelogram? Opposite sides of a parallelogram are congruent. Opposite angles of a parallelogram are congruent. The diagonals of a parallelogram bisect each other.
SummarizeHave students make a graphic organizer to
summarize what they know about the sides, angles,
and diagonals of a parallelogram. A sample is
shown below.
Exercise 1: Students practice what they learned
in part 2 of the lesson.
Exercise 2: Students use reasoning to extend what
they know about parallelograms.
Exercise 3: Students use reasoning and/or algebra
to find unknown angle measures.
Exercise 4: Students apply their learning to solve
a multi-step real-world problem.
CLOSE
PRACTICE
Highlighting the Standards
Exercise 4 is a multi-part exercise that includes
opportunities for mathematical modeling,
reasoning, and communication. It is a good
opportunity to address Standard 4 (Model with
mathematics). Draw students’ attention to the
way they interpret their mathematical results
in the context of the real-world situation.
Specifically, ask students to explain what their
mathematical findings tell them about the
appearance and layout of the park.
A B
CD P
x˚x˚ y˚
y˚
z˚
A B
CD
P
A B
CD
P
A B
CD
P
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4. A city planner is designing a park in the shape of a
parallelogram. As shown in the figure, there will be
two straight paths through which visitors may enter
the park. The paths are bisectors of consecutive angles
of the parallelogram, and the paths intersect at point P.
a. Work directly on the parallelograms below and use a compass and
straightedge to construct the bisectors of ∠A and ∠B. Then use a protractor to
measure ∠APB in each case.
Make a conjecture about ∠APB.
b. Write a paragraph proof to show that your conjecture is always true. (Hint: Suppose m∠BAP = x°, m∠ABP = y°, and m∠APB = z°. What do you know
about x + y + z? What do you know about 2x + 2y?)
c. When the city planner takes into account the
dimensions of the park, she finds that point P lies on ___ DC , as shown. Explain why it must be the case that
DC = 2AD. (Hint: Use congruent base angles to show
that �DAP and �CPB are isosceles.)
By the Triangle Sum Theorem, x + y + z = 180. Also, m∠DAB = (2x)° and
m∠ABC = (2y)°. By the Same-Side Interior Angles Postulate m∠DAB +
m∠ABC = 180°. So 2x + 2y = 180 and x + y = 90. Substituting this in the
first equation gives 90 + z = 180 and z = 90.
∠DAP � ∠BAP since ___
AP is an angle bisector. Also, ∠DPA � ∠BAP by the
Alternate Interior Angles Theorem. Therefore, ∠DAP � ∠DPA. This means
�DAP is isosceles, with ___
AD � ___
DP . Similarly, ___
BC � ___
PC . Also, ___
BC � ___
AD as
opposite sides of a parallelogram. So, DC = DP + PC = AD + BC = AD +
AD = 2AD.
∠APB is a right angle.
Chapter 6 250 Lesson 2
A B
D C
J K N1 3
254
43˚62˚
M L
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P R A C T I C E
The angles of a parallelogram also have an important property. It is stated in the
following theorem, which you will prove as an exercise.
1. Prove the above theorem about opposite angles of a parallelogram.
Given: ABCD is a parallelogram.
Prove: ∠A � ∠C and ∠B � ∠D
(Hint: You only need to prove that ∠A � ∠C. A similar
argument can be used to prove that ∠B � ∠D. Also,
you may or may not need to use all the rows of the table
in your proof.)
2. Explain why consecutive angles of a parallelogram are supplementary.
3. In the figure, JKLM is a parallelogram. Find the measure of each of the
numbered angles.
Statements Reasons
1. 1.
2. 2.
3. 3.
4. 4.
5. 5.
6. 6.
7. 7.
Theorem
If a quadrilateral is a parallelogram, then opposite angles are congruent.
T
ABCD is a parallelogram.
Draw ___
DB .
___
AB || ___
DC ; ___
AD || ___
BC
∠ADB � ∠CBD; ∠ABD � ∠CDB
___
DB � ___
DB
�ABD � �CDB
∠A � ∠C
Given
Through any two points there exists exactly one line.
Definition of parallelogram
ASA Congruence Criterion
CPCTC
Alternate Interior Angles Theorem
Reflexive Property of Congruence
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.
m∠1 = 19°; m∠2 = 43°; m∠3 = 118°;
m∠4 = 118°; m∠5 = 19°
Chapter 6 249 Lesson 2
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Chapter 6 250 Lesson 2
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Notes
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Assign these pages to help your students practice and apply important lesson concepts. For additional exercises, see the Student Edition.
Answers
Additional Practice
1. 108.8 cm 2. 91 cm
3. 217.6 cm 4. 123°
5. 123° 6. 57°
7. 117° 8. 63°
9. 71 10. 21
11. 10.5 12. 15
13. 30 14. (0, -3)
15. Possible answer:
Problem Solving
1. m∠C = 135°; m∠D = 45°
2. 15 in. 3. 4.5 ft
4. 65° 5. B
6. H 7. D
Statements Reasons1. DEFG is a parallelogram. 1. Given
2. m∠EDG = m∠EDH + m∠GDH, m∠FGD = m∠FGH + m∠DGH
2. Angle Add. Post.
3. m∠EDG + m∠FGD = 180° 3. � � cons. ∠ supp.
4. m∠EDH + m∠GDH + m∠FGH + m∠DGH = 180°
4. Subst. (Steps 2, 3)
5. m∠GDH + m∠DGH + m∠DHG = 180°
5. Triangle Sum Thm.
6. m∠GDH + m∠DGH + m∠DHG = m∠EDH + m∠GDH + m∠FGH + m∠DGH
6. Trans. Prop. of =
7. m∠DHG = m∠EDH + m∠FGH
7. Subtr. Prop. of =
s
ADDITIONAL PRACTICE AND PROBLEM SOLVING
Chapter 6 251 Lesson 2
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Problem Solving
Chapter 6 252 Lesson 2
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6-2Name Class Date
Additional Practice
Chapter 6 251 Lesson 2
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Notes
Chapter 6 252 Lesson 2