Standards for TEACH Mathematical Content - Middle School · PDF fileMathematical Content ......

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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


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.


G_MFLBESE200852_C06L02.indd 245 03/05/14 4:56 PM

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Notes

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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.


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


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.


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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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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.


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.


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°


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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


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Problem Solving


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6-2Name Class Date

Additional Practice


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Notes


Standards for TEACH Mathematical Content - Middle School · PDF fileMathematical Content ......

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Transcript of Standards for TEACH Mathematical Content - Middle School · PDF fileMathematical Content ......