CorrectionKey=NL-B;CA-B CorrectionKey=NL-C;CA … of the Alternate Interior Angles Theorem 6. 7. ......

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Explore Proving the Opposite Sides Criterion for a Parallelogram

You can prove that a quadrilateral is a parallelogram by using the definition of a parallelogram. That is, you can show that both pairs of opposite sides are parallel. However, there are other conditions that also guarantee that a quadrilateral is a parallelogram.

Theorem

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Complete the proof of the theorem.

Given: _ AB ≅ _ CD and

_ AD ≅ _ CB

Prove: ABCD is a parallelogram.

A Draw diagonal _ DB .

Why is it helpful to draw this diagonal?

B Use triangle congruence theorems and corresponding parts to complete the proof that the opposite sides are parallel so the quadrilateral is a parallelogram.

Statements Reasons

1. Draw _ DB . 1. Through any two points, there is exactly one line.

2. _ DB ≅ _ DB 2.

3. _ AB ≅ _ CD ; _ AD ≅

_ CB 3.

4. △ABD ≅ △CDB 4.

5. ∠ABD ≅ ∠CDB; ∠ADB ≅ ∠CBD 5.

6. _ AB ‖ _ DC ; _ AD ‖ _ BC 6.

7. ABCD is a parallelogram. 7.

Resource Locker

D C

BA

Drawing the auxiliary line, diagonal ― DB , creates two triangles to use in the proof.

Reflexive Property of Congruence

Given

SSS Triangle Congruence Theorem

CPCTC

Converse of the Alternate Interior Angles Theorem

Definition of parallelogram

Module 24 1203 Lesson 2

24.2 Conditions for ParallelogramsEssential Question: What criteria can you use to prove that a quadrilateral is a parallelogram?

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Common Core Math StandardsThe student is expected to:

G-CO.11

Prove theorems about parallelograms. Also G-CO.10, G-SRT.5

Mathematical Practices

MP.2 Reasoning

Language ObjectiveExplain to a partner how to identify the opposite sides, opposite angles, and consecutive angles and sides of a quadrilateral.

HARDCOVER PAGES 975986

Turn to these pages to find this lesson in the hardcover student edition.

Conditions for Parallelograms

ENGAGE Essential Question: What criteria can you use to prove that a quadrilateral is a parallelogram?Opposite sides are congruent; opposite sides are

parallel; opposite angles are congruent; one angle is

supplementary to both its consecutive angles; a pair

of opposite sides are congruent and parallel;

diagonals bisect one another.

PREVIEW: LESSON PERFORMANCE TASKView the Engage section online. Discuss the photo. Be sure students understand that the stars in a constellation are at various distances from Earth, and that the shape the constellation appears to take is purely a line-of-sight effect. Then preview the Lesson Performance Task.

1203

HARDCOVER

Turn to these pages to find this lesson in the hardcover student edition.

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Explore Proving the Opposite Sides Criterion

for a Parallelogram

You can prove that a quadrilateral is a parallelogram by using the definition of a parallelogram.

That is, you can show that both pairs of opposite sides are parallel. However, there are other

conditions that also guarantee that a quadrilateral is a parallelogram.

Theorem

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral

is a parallelogram.

Complete the proof of the theorem.

Given: _ AB ≅

_ CD and _ AD ≅

_ CB


Draw diagonal _ DB .

Why is it helpful to draw this diagonal?

Use triangle congruence theorems and corresponding parts to complete the proof that the

opposite sides are parallel so the quadrilateral is a parallelogram.

Statements

Reasons

1. Draw _

DB .

1. Through any two points, there is exactly one line.

2. _

DB ≅ _

DB

2.

3. _

AB ≅ _

CD ; _ AD ≅

_ CB

3.

4. △ _ ABD ≅ △

_ CDB 4.

5. ∠ABD ≅ ∠CDB; ∠ADB ≅ ∠CBD 5.

6. _

AB ‖ _

DC ; _ AD ‖

_ BC

6.


Resource

Locker

G-CO.11 Prove theorems about parallelograms. Also G-CO.10, G-SRT.5

DC

BA

Drawing the auxiliary line, diagonal ― DB , creates two triangles to use in the proof.

Reflexive Property of Congruence

Given


CPCTC



Module 24

1203

Lesson 2

24 . 2 Conditions for Parallelograms

Essential Question: What criteria can you use to prove that a quadrilateral is a parallelogram?

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1203 Lesson 24 . 2

L E S S O N 24 . 2

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It is possible to combine the theorem from the Explore and the definition of a parallelogram to state the following condition for proving a quadrilateral is a parallelogram. You will prove this in the exercises.

Theorem

If one pair of opposite sides of a quadrilateral are parallel and congruent, then the quadrilateral is a parallelogram.

Reflect

1. Discussion A quadrilateral has two sides that are 3 cm long and two sides that are 5 cm long. A student states that the quadrilateral must be a parallelogram. Do you agree? Explain.

Explain 1 Proving the Opposite Angles Criterion for a Parallelogram

You can use relationships between angles to prove that a quadrilateral is a parallelogram.

Theorem

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Example 1 Prove that a quadrilateral is a parallelogram if its opposite angles are congruent.

Given: ∠A ≅ ∠C and ∠B ≅ ∠D Prove: ABCD is a parallelogram.

m∠A + m∠B + m∠C + m∠D = 360° by .

From the given information, m∠A = m∠ and m∠B = m∠ . By substitution,

m∠A + m∠D + m∠A + m∠D = 360° or 2m∠ + 2m∠ = 360°. Dividing

both sides by 2 gives . Therefore, ∠A and ∠D are

supplementary and so _ AB ‖ _ DC by the

A similar argument shows that _ AD ‖ _ BC , so ABCD is a parallelogram

by .

Reflect

2. What property or theorem justifies dividing both sides of the equation by 2 in the above proof?

D C

BA

Disagree; you can only conclude that the quadrilateral is a parallelogram if you know that

the congruent sides are opposite each other.

the Polygon Angle Sum Theorem

C

A D

D

the definition of parallelogram

Division Property of Equality

Converse of the Same-Side Interior Angles Postulate.

m∠A + m∠D = 180°


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IN1_MNLESE389762_U9M24L2 1204 9/30/14 12:23 PM

Math BackgroundIn this lesson, students extend their earlier work with the properties of parallelograms to establish sufficient conditions for concluding that a quadrilateral is a parallelogram. A quadrilateral is a parallelogram if its opposite sides are parallel, if its opposite sides are congruent, if its opposite angles are congruent, if its diagonals bisect each other, or if one pair of opposite sides is congruent and parallel. It is worth emphasizing that the statements to prove in the previous lesson all have the form “If a quadrilateral is a parallelogram, then [property].” In contrast, the statements to prove in this lesson all have the form, “If [condition], then the quadrilateral is a parallelogram.”

EXPLORE Proving the Opposite Sides Criterion for a Parallelogram

INTEGRATE TECHNOLOGYStudents have the option of doing the proof activity either in the book or online.

QUESTIONING STRATEGIESWhat is a sufficient condition to prove that a quadrilateral is a parallelogram? Either both

pairs of opposite sides are parallel, or both pairs of

opposite sides are congruent.

EXPLAIN 1 Proving the Opposite Angles Criterion for a Parallelogram

INTEGRATE MATHEMATICS PRACTICESFocus on Math ConnectionsMP.1 The proof of this criterion depends on students understanding the hypothesis and conclusion of the statement. Both pairs of opposite angles are given to be congruent, and that is part of the hypothesis. Having these angles congruent leads to both pairs of opposite sides of the quadrilateral being parallel, which satisfies the definition of a parallelogram. Stating that the quadrilateral is a parallelogram is the conclusion.

QUESTIONING STRATEGIESSuppose only one pair of opposite angles of a quadrilateral are congruent. Can you still

conclude that the quadrilateral is a parallelogram? Explain. No; in this case, the quadrilateral could

be a kite.

PROFESSIONAL DEVELOPMENT

Conditions for Parallelograms 1204


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Explain 2 Proving the Bisecting Diagonals Criterion for a Parallelogram

You can use information about the diagonals in a given figure to show that the figure is a parallelogram.

Theorem

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

Example 2 Prove that a quadrilateral whose diagonals bisect each other is a parallelogram.

Given: _ AE ≅ _ CE and

_ DE ≅

_ BE


Statements Reasons

1. _ AE ≅

_ CE , _ DE ≅

_ BE 1.

2. ∠AEB ≅ ∠CED, ∠AED ≅ ∠CEB 2.

3. 3.

4. _ AB ≅ _ CD , _ AD ≅

_ CB 4.


Reflect

3. Critique Reasoning A student claimed that you can also write the proof using the SSS Triangle Congruence Theorem since ― AB ≅ ― CD and ― AD ≅ ― CB . Do you agree? Justify your response.

Explain 3 Using a Parallelogram to Prove the Concurrency of the Medians of a Triangle

Sometimes properties of one type of geometric figure can be used to recognize properties of another geometric figure. Recall that you explored triangles and found that the medians of a triangle are concurrent at a point that is 2 __ 3 of the distance from each vertex to the midpoint of the opposite side. You can prove this theorem using one of the conditions for a parallelogram from this lesson.

Example 3 Complete the proof of the Concurrency of Medians of a Triangle Theorem.

Given: △ABC

Prove: The medians of △ABC are concurrent at a point that is 2 __ 3 of the distance from each vertex to the midpoint of the opposite side.

D C

BE

A

If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram.

Vertical angles are congruent.

Given

SAS Triangle Congruence Theorem

CPCTC

△AEB ≅ △CED, △AED ≅ △CEB

No; in order to use the fact that opposite sides are congruent, you must first know that the

quadrilateral is a parallelogram.


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IN1_MNLESE389762_U9M24L2.indd 1205 4/18/14 7:54 PM

COLLABORATIVE LEARNING

Peer-to-Peer ActivityDivide students into pairs. Have one student draw different quadrilaterals (kites, rhombuses, rectangles, etc.). Have the other student determine whether the quadrilateral meets any of the criteria for a parallelogram, and list them. After a while, have the students switch roles.

EXPLAIN 2 Proving the Bisecting Diagonals Criterion for a Parallelogram

INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Students should read this criterion carefully and pay attention to the sufficient conditions given and to how the criterion is proved. The proof depends on students using the SAS Congruence Criterion, the Vertical Angles Theorem, and the Opposite Sides Criterion. Pairs of triangles formed by the diagonals are congruent, and this makes the opposite sides of the quadrilateral congruent, so that the Opposite Sides Criterion (for a parallelogram) applies.

QUESTIONING STRATEGIESHow does the proof of this criterion use the Opposite Sides Criterion? The proof uses the

SAS triangle congruence criteria and the fact that

corresponding parts of congruent triangles are

congruent to show that the opposite sides of

a quadrilateral are congruent. Then the Opposite

Sides Criterion is applied to state that the

quadrilateral must be a parallelogram.

In the proof of the Opposite Sides Criterion, which theorem justifies that both pairs of

opposite sides are parallel? Once the alternate

interior angles are stated to be congruent, the


states that the lines are parallel.

1205 Lesson 24 . 2


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Let △ABC be a triangle such that M is the midpoint of _ AB and N is the midpoint

of _ BC . Label the point where the two medians intersect as P. Draw

_ MN .

_ MN is a midsegment of △ABC because it connects the

midpoints of two sides of the triangle.

_ MN is parallel to and MN = by the

Triangle Midsegment Theorem.

Let Q be the midpoint of _ PA and let R be the midpoint of

_ PC .

Draw _ QR .

_ QR is a midsegment of △APC because it connects the

midpoints of two sides of the triangle.

_ QR is parallel to and QR = by the

Triangle Midsegment Theorem.

So, you can conclude that MN = by substitution and that

― MN ‖ ― QR because .

Now draw _ MQ and

_ NR and consider quadrilateral MQRN.

Quadrilateral MQRN is a parallelogram because

.

Since the diagonals of a parallelogram bisect each other, then QP = .

Also, AQ = QP since .

Therefore, AQ = QP = . This shows that point P is located on _ AN at a point that is 2 __ 3 of the

distance from A to N.

By similar reasoning, the diagonals of a parallelogram bisect each other, so RP = .

Also, CR = RP since .

Therefore, CR = RP = . This shows that point P is located on _ CM at a point that is 2 __ 3 of the

distance from C to M.

You can repeat the proof using any two medians of △ABC. The same reasoning shows that the medians from vertices B and C intersect at a point that is also 2 __ 3 of the distance from C to M, so this point must also be point P. This shows that the three medians intersect at a unique point P and that the point is 2 __ 3 of the distance from each vertex to the midpoint of the opposite side.

M

CB

A

N

P

M

CB

A

N

P

Q

R

M

CB

A

N

P

Q

R

― AC

― AC

QR

1 __ 2 AC

1 __ 2 AC

both segments are parallel to the same segment

a pair of opposite sides are congruent and parallel

Q is the midpoint of ― PA

R is the midpoint of ― PC

NP

NP

MP

MP


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EXPLAIN 3 Using a Parallelogram to Prove the Concurrency of the Medians of a Triangle

QUESTIONING STRATEGIESHow is the Triangle Midsegment Theorem used in proving the concurrency of the

medians in a triangle? The Triangle Midsegment

Theorem is used to show that the opposite sides of

a quadrilateral are parallel. This creates a

parallelogram within the triangle. Then, since the

diagonals of a parallelogram bisect each other, the

parallelogram in turn is used to prove the

concurrency of the medians in the triangle.

AVOID COMMON ERRORSStudents may be confused when asked to locate the midsegments in a triangle that already has the medians labeled for this proof. Have students use one color pencil to mark the medians, a second color pencil to mark the midsegments, and a third color pencil to mark the congruent segments on the diagonals of the parallelogram.

DIFFERENTIATE INSTRUCTION

Kinesthetic ExperienceHave students use raw spaghetti to demonstrate the theorems in the lesson. For example, ask them to try to form a parallelogram with opposite sides that are congruent but not parallel, or vice versa. This emphasizes that, in the statement that a quadrilateral with a pair of opposite sides parallel and congruent leads to a parallelogram, the same pair of opposite sides must be congruent and parallel.



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Reflect

4. In the proof, how do you know that point P is located on _ AN at a point that is 2 __ 3 of the

distance from A to N?

Explain 4 Verifying Figures Are ParallelogramsYou can use information about sides, angles, and diagonals in a given figure to show that the figure is a parallelogram.

Example 4 Show that each quadrilateral is a parallelogram for the given values of the variables.

x = 7 and y = 4

Step 1 Find BC and DA.

BC = x + 14 = 7 + 14 = 21

DA = 3x = 3 (7) = 21

Step 2 Find AB and CD.

AB = 5y - 4 = 5 (4) - 4 = 16

CD = 2y + 8 = 2 (4) + 8 = 16

So, BC = DA and AB = CD. ABCD is a parallelogram since both pairs of opposite sides are congruent.

z = 11 and w = 4.5

Step 1 Find m∠F and m∠H.

m∠F = =

m∠H = =

Step 2 Find m∠E and m∠G.

m∠E = =

m∠G = =

So, m∠F = m∠ and m∠E = m∠ . EFGH is a parallelogram since

Reflect

5. What conclusions can you make about _ FG and

_ EH in Part B? Explain.

D

CB

A

5y - 4 2y + 8

x + 14

3x

H

GF

E

(9z + 19)° (14w - 1)°

(11z - 3)°(12w + 8)°

Possible answer: Since AQ = QP = PN, ― AN is divided into three segments of equal

length (AQ = QP = PN) . So, points P and Q divide ― AN into thirds. Therefore, since

AP = AQ + QP = 1 __ 3 AN + 1 __ 3 AN = 2 __ 3 AN, then point P is 2 __ 3 of the distance from A to N.

(9z + 19) °

(12w + 8) °

(11z - 3) °

(9 (11) + 19) ° = 118°

both pairs of opposite angles are congruent.

(11 (11) - 3) ° = 118°

(12 (4.5) + 8) ° = 62°

(14w - 1) ° (14 (4.5) - 1) ° = 62°

H G

― FG is parallel and congruent to ― EH ; EFGH is a parallelogram and opposite sides of a

parallelogram are parallel and congruent.



IN1_MNLESE389762_U9M24L2 1207 10/16/14 2:53 PM

LANGUAGE SUPPORT

Connect VocabularyEncourage students to pay close attention to the key words in each theorem, such as opposite, parallel, and congruent. Have them write each of the theorems on note cards, illustrate the theorems, and highlight these key words.

EXPLAIN 4 Verifying Figures Are Parallelograms

QUESTIONING STRATEGIESHow is the Opposite Sides Criterion used to show that the given quadrilateral is a

parallelogram? Algebra is used to show that the

opposite sides are congruent after the given values

are substituted into the expression. The figure

meets the criterion and must be a parallelogram.

AVOID COMMON ERRORSSome students may not know which criterion to use when proving that a quadrilateral is a parallelogram. Explain that they should study the given information carefully to determine which condition is best to apply. For example, if they are given congruent angles, then they should look only at the two criteria that deal with angles: both pairs of opposite angles are congruent, or one angle is supplementary to both of its consecutive angles.

1207 Lesson 24 . 2


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Your Turn

Show that each quadrilateral is a parallelogram for the given values of the variables.

6. a = 2.4 and b = 9 7. x = 6 and y = 3.5

Elaborate

8. How are the theorems in this lesson different from the theorems in the previous lesson, Properties of Parallelograms?

9. Why is the proof of the Concurrency of the Medians of a Triangle Theorem in this lesson and not in the earlier module when the theorem was first introduced?

10. Essential Question Check-In Describe three different ways to show that quadrilateral ABCD is a parallelogram.

S

RQ

P

(10b - 16)° (9b + 25)°2a + 127a

M L

KJ

N

2x + 2

6y + 1 4x - 10

8y - 6

A B

C D

PQ = 7a = 7 (2.4) = 16.8

RS = 2a + 12 = 2 (2.4) + 12 = 16.8

m∠Q = (10b - 16) ° = (10 (9) - 16) ° = 74°

m∠R = (9b + 25) ° = (9 (9) + 25) ° = 106°

So, PQ = RS. Also, ― PQ ‖ ― RS since same-side

interior angles are supplementary.

PQRS is a parallelogram since a pair of

opposite sides are parallel and congruent.

JN = 2x + 2 = 2 (6) + 2 = 14

LN = 4x - 10 = 4 (6) - 10 = 14

KN = 8y - 6 = 8 (3.5) - 6 = 22

MN = 6y + 1 = 6 (3.5) + 1 = 22

So JN = LN and KN = MN. JKLM is a

parallelogram since the diagonals

bisect each other.

In the previous lesson, you begin with a quadrilateral that is known to be a parallelogram

and each theorem states a property of the parallelogram. In this lesson, you begin with a

quadrilateral and each theorem states conditions that guarantee that the quadrilateral is

a parallelogram.

The proof involves creating a quadrilateral that needs to be shown is a parallelogram.

This could not be done before this lesson.

Possible answer: Since right angles are congruent, opposite angles of ABCD are congruent.

Therefore, ABCD is a parallelogram. Opposite sides of ABCD are congruent, so ABCD is a

parallelogram. ∠A is supplementary to ∠D, so ― AB ‖ ― CD . Therefore, ABCD is a parallelogram

since a pair of opposite sides are congruent and parallel.





ELABORATE

QUESTIONING STRATEGIESHow do the theorems in this lesson differ from the ones in the previous lesson? They

are converses. Here we know the properties, and are

proving the quadrilateral is a parallelogram;

previously, we knew the figure was a parallelogram

and we were proving properties.

SUMMARIZE THE LESSONTwo opposite angles in a quadrilateral each measure 103°. Can you conclude that this

quadrilateral is a parallelogram? Explain why or draw a counterexample. No, we can’t, because only one

pair of opposite angles is congruent. Possible

counterexample:



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Evaluate: Homework and Practice

1. You have seen a proof that if both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Write the proof as a flow proof.


_ AD ≅ _ CB


• Online Homework• Hints and Help• Extra Practice

D C

BA

AB ‖ DC, AD ‖ BC

Given Reflexive Propertyof Congruence

Given

Through any two points,there is exactly one line.

DB ≅ DBAD ≅ CBAB ≅ CD

▵ABD ≅ ▵CDB


Converse of the Alternate InteriorAngles Theorem

CPCTC

ABCD is aparallelogram.


∠ADB ≅ ∠CBD,∠ABD ≅ ∠CDB

Draw DB.

Possible answer:



IN1_MNLESE389762_U9M24L2 1209 9/30/14 12:31 PMExercise Depth of Knowledge (D.O.K.) Mathematical Practices

1–4 2 Skills/Concepts MP.3 Logic

5–12 1 Recall MP.5 Using Tools

13–18 2 Skills/Concepts MP.2 Reasoning

19 3 Strategic Thinking MP.3 Logic

20 3 Strategic Thinking MP.4 Modeling

21 3 Strategic Thinking MP.2 Reasoning

EVALUATE

ASSIGNMENT GUIDE

Concepts and Skills Practice

ExploreProving the Opposite Sides Criterion for a Parallelogram

Exercise 1

Example 1Proving the Opposite Angles Criterion for a Parallelogram

Exercise 2

Example 2Proving the Bisecting Diagonals Criterion for a Parallelogram

Exercise 3

Example 3Using a Parallelogram to Prove the Concurrency of the Medians of a Triangle

Exercise 4

Example 4Verifying Figures are Parallelograms

Exercises 5–12

INTEGRATE MATHEMATICAL PRACTICESFocus on ModelingMP.4 Some students may benefit from a hands-on approach for exploring which quadrilaterals are parallelograms. Have students copy a larger version of the quadrilateral in an exercise onto a sheet of paper and then cut it out. Have them use a ruler to verify that the opposite sides are congruent, or a protractor to verify that the opposite angles are congruent, so that the criteria for the quadrilateral to be a parallelogram are satisfied.

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2. You have seen a proof that if both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Write the proof as a two-column proof.

Given: ∠A ≅ ∠C and ∠B ≅ ∠D Prove: ABCD is a parallelogram.

Statements Reasons

1. 1.

3. You have seen a proof that if the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Write the proof as a paragraph proof.

Given: _ AE ≅ _ CE and

_ DE ≅

_ BE


4. Complete the following proof of the Triangle Midsegment Theorem.

Given: D is the midpoint of _ AC , and E is the midpoint of

_ BC .

Prove: _ DE ‖ _ AB , DE = 1 __ 2 AB

Extend _ DE to form

_ DF such that

_ DE ≅

_ FE . Then draw

_ BF , as shown.

D C

BA

C

E FD

A B

D C

BE

A

C

ED

A B

Possible answer: It is given that ― AE ≅

― CE and ― DE ≅

― BE . Since vertical angles are congruent, ∠AEB ≅ ∠CED and ∠AED ≅ ∠CEB. Therefore, △AEB ≅ △CED and △AED ≅ △CEB by the SAS Triangle Congruence Theorem. Since corresponding parts of congruent triangles are congruent,

― AB ≅ ― CD and

― AD ≅ ― CB . So, ABCD is a parallelogram because if both pairs of

opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

∠A ≅ ∠C and ∠B ≅ ∠D Given

2. m∠A + m∠B + m∠C + m∠D = 360° 2. Polygon Angle Sum Theorem

3. m∠A + m∠B + m∠A + m∠B = 360°; m∠A + m∠D + m∠A + m∠D = 360° 3. Substitution Property of Equality

4. 2m∠A + 2m∠B = 360°; 2m∠A + 2m∠D = 360° 4. Distributive Property

5. m∠A + m∠B = 180°; m∠A + m∠D = 180° 5. Division Property of Equality

6. ∠A and ∠B are supplementary; ∠A and ∠D are supplementary.

6. Definition of supplementary angles

7. ― AD ‖

― BC and ― AB ‖

― DC 7. Converse of the Same-Side Interior

Angles Postulate

8. ABCD is a parallelogram. 8. Definition of parallelogram





INTEGRATE MATHEMATICAL PRACTICESFocus on ReasoningMP.2 Point out that only one criterion of the six listed below needs to be met to show that a quadrilateral is a parallelogram: Both pairs of opposite sides are parallel. One pair of opposite sides is parallel and congruent. 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.



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It is given that E is the midpoint of _ CB , so

_ CE ≅ .

By the Vertical Angles Theorem, ∠CED ≅ .

So, △CED ≅ by .

Since corresponding parts of congruent triangles are congruent, _ CD ≅ .

D is the midpoint of _ AC , so

_ CD ≅ .

By the Transitive Property of Congruence, _ AD ≅ .

Also, since corresponding parts of congruent triangles are congruent, ∠CDE ≅ .

So, _ AC ‖ _ FB by .

This shows that DFBA is a parallelogram because .

By the definition of parallelogram, _ DE is parallel to .

Since opposite sides of a parallelogram are congruent, AB = .

_ DE ≅

_ FE , so DE = 1 __ 2 and by substitution, DE = 1 __ 2 .

Show that each quadrilateral is a parallelogram for the given values of the variables.

5. x = 4 and y = 9 6. u = 8 and v = 3.5

Determine if each quadrilateral must be a parallelogram. Justify your answer.

7. 8.

J

My + 4 2y - 5

5x + 7

6x + 3 L

K

A2u + 3

4v - 36v - 10

3u - 5C

B

D

E

― BE

∠BEF

∠BFE

DF AB

― AD

― BF

― BF

the SAS Triangle Congruence Theorem


a pair of opposite sides

△BEF

― AB

FD

are congruent and parallel

JK = 5x + 7 = 5 (4) + 7 = 27

LM = 6x + 3 = 6 (4) + 3 = 27

KL = 2y − 5 = 2 (9) − 5 = 13

MJ = y + 4 = 9 + 4 = 13

So, JK = LM and KL = MJ. JKLM is a parallelogram.

AE = 2u + 3 = 2(8) + 3 = 19

CE = 3u − 5 = 3(8) − 5 = 19

BE = 4v − 3 = 4(3.5) − 3 = 11

DE = 6v − 10 = 6(3.5) − 10 = 11

So AE = CE and BE = DE. ABCD is a parallelogram.

No. One pair of opposite sides are parallel. A different pair of opposite sides are congruent.

Yes. the third pair of angles in the triangles are congruent. So, both pairs of opposite angles are congruent.




1211 Lesson 24 . 2


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9. 10.

11. 12.

13. Communicate Mathematical Ideas Kalil wants to write the proof that the medians of a triangle are concurrent at a point that is 2 __ 3 of the distance from each vertex to the midpoint of the opposite side. He starts by drawing △PQR and two medians,

_ PK and

_ QL . He labels the point of intersection as

point J, as shown. What segment should Kalil draw next? What conclusions can he make about this segment? Explain.

14. Critical Thinking Jasmina said that you can draw a parallelogram using the following steps.

1. Draw a point P.

2. Use a ruler to draw a segment that is 1 inch long with its midpoint at P.

3. Use the ruler to draw a segment that is 2 inches long with its midpoint at P.

4. Use the ruler to connect the endpoints of the segments to form a parallelogram.

Does Jasmina’s method always work? Is there ever a time when it would not produce a parallelogram? Explain.

107°107°

73°

57° 57°123°

123°

K

Q

L

J

R

P

Yes. A pair of alternate interior angles are congruent, so a pair of opposite sides are parallel. The same pair of opposite sides are congruent by SAS and CPCTC.

No. One pair of opposite sides are congruent and one diagonal is bisected by the other.

Yes. The 73° angle is supplementary to both of the 107° angles. This shows that both pairs of opposite sides are parallel by the Converse of the Same-Side Interior Angles Postulate.

No. You are only given the measures of the four angles formed by the intersecting diagonals of the quadrilateral.

He should draw _ KL . _ KL is a midsegment of △PQR because it connects the

midpoints of two sides of the triangle. ― KL is parallel to

― PQ and KL = 1 _ 2 PQ

by the Triangle Midsegment Theorem.

The method always works as long as the second segment does not overlap the first segment. The two segments are diagonals. Since the diagonals bisect each other, the quadrilateral is a parallelogram.





PEERTOPEER DISCUSSIONAsk students to discuss with a partner how to create a graphic organizer that will display all the criteria they have learned that make a quadrilateral a parallelogram. Then have them create the graphic organizer and make sure they include the following criteria:

• Opposite sides parallel

• Opposite sides congruent

• Opposite angles congruent

• Consecutive angles supplementary

• Diagonals bisecting each other

• One pair of sides congruent and parallel



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15. Critique Reasoning Matthew said that there is another condition for parallelograms. He said that if a quadrilateral has two congruent diagonals, then the quadrilateral is a parallelogram. Do you agree? If so, explain why. If not, give a counterexample to show why the condition does not work.

16. A parallel rule can be used to plot a course on a navigation chart. The tool is made of two rulers connected at hinges to two congruent crossbars,

_ AD

and _ BC . You place the edge of one ruler on your desired course and then

move the second ruler over the compass rose on the chart to read the bearing for your course. If

_ AD ‖ _ BC , why is

_ AB always parallel to

_ CD ?

17. Write a two-column proof to prove that a quadrilateral with a pair of opposite sides that are parallel and congruent is a parallelogram.


_ AB ‖ _ CD

Prove: ABCD is a parallelogram. (Hint: Draw _ DB .)

Statements Reasons

1. 1.

A B

CD

D C

BA

No. The figure shows a counterexample. The diagonals are congruent, but the quadrilateral does not have two pairs of parallel opposite sides, so it is not a parallelogram.

Possible answer: ― AD and

― BC are opposite sides of a quadrilateral. It is given that

― AD ‖ _ BC and

― AD ≅ ― BC . Therefore, ABCD is a

parallelogram. So, ― AB is always parallel to

― CD .

Draw ― DB Through any two

points, there is exactly one line.

2. ― AB ≅ ― CD 2. Given

3. ― AB ‖ ― CD 3. Given

4. ∠ABD ≅ ∠CDB 4. Alternate Interior Angles Theorem

5. ― DB ≅ ― DB 5. Reflexive Property of Congruence

6. △ABD ≅ △CDB 6. SAS Triangle Congruence Theorem

7. ― AD ≅ ― CB 7. CPCTC

8. ABCD is a parallelogram. 8. If both pairs of opposite sides of a

quadrilateral are congruent, then the quadrilateral is a parallelogram.


DO NOT EDIT--Changes must be made through “File info” CorrectionKey=NL-B;CA-B

IN1_MNLESE389762_U9M24L2 1213 9/30/14 12:32 PM

1213 Lesson 24 . 2


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18. Does each set of given information guarantee that quadrilateral JKLM is a parallelogram? Select the correct answer for each lettered part.

A. JN = 25 cm, JL = 50 cm, KN = 13 cm, KM = 26 cm Yes No

B. ∠MJL ≅ ∠KLJ, _ JM ≅ _ LK Yes No

C. _ JM ≅ _ JK , _ KL ≅ _ LM Yes No

D. ∠MJL ≅ ∠MLJ, ∠KJL ≅ ∠KLJ Yes No

E. △JKN ≅ △LMN Yes No

H.O.T. Focus on Higher Order Thinking

19. Explain the Error A student wrote the two-column proof below. Explain the student’s error and explain how to write the proof correctly.

Given: ∠1 ≅ ∠2, E is the midpoint of _ AC .


Statements Reasons

1. ∠1 ≅ ∠2 1. Given

2. E is the midpoint of _ AC . 2. Given

3. _ AE ≅

_ CE 3. Definition of midpoint

4. ∠AED ≅ ∠CEB 4. Vertical angles are congruent.

5. △AED ≅ △CEB 5. ASA Triangle Congruence Theorem

6. _ AD ≅

_ CB 6. Corresponding parts of congruent

triangles are congruent

7. ABCD is a parallelogram. 7. If a pair of opposite sides of a quadrillateral are congruent, then the quadrillateral is a parallelogram.

K

M L

JN

D

BA

C

E2

1

a. Yes; The diagonals of the quadrilateral bisect each other, so JKLM is a parallelogram.

b. Yes; A pair of opposite sides are both parallel and congruent, so JKLM is a parallelogram.

c. No; none of the conditions for a parallelogram are met.

d. No; none of the conditions for a parallelogram are met.

e. Yes; A pair of opposite sides are both parallel and congruent, so JKLM is a parallelogram.

Possible answer: The student used an invalid reason in Step 7. The student should also show that ̄ AD ‖ ̄ CB . This is true because ∠1 ≅ ∠2 and these are alternate interior angles. Then the student can conclude that ABCD is a parallelogram because a pair of opposite sides are both parallel and congruent.





INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 At this point, students may be thinking of the theorems from Lesson 9–1 and Lesson 9–2 as “if and only if statements” (biconditionals) about the relationship between quadrilaterals and parallelograms. Many of the theorems can be combined to be biconditionals, but make sure students can still focus on which phrase is the hypothesis and which phrase is the conclusion when they apply the theorems. For the exercises in this lesson, the hypothesis includes a statement that a quadrilateral has certain given attributes. The conclusion of the statement is that the quadrilateral is a parallelogram.



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20. Persevere in Problem Solving The plan for a city park shows that the park is a quadrilateral with straight paths along the diagonals. For what values of the variables is the park a parallelogram? In this case, what are the lengths of the paths?

21. Analyze Relationships When you connect the midpoints of the consecutive sides of any quadrilateral, the resulting quadrilateral is a parallelogram. Use the figure below to explain why this is true. (Hint: Draw a diagonal of ABCD.)

T

V

(2x + 4y) m

(16y - 10) m

(12y + 50) m SR

U

(5x + 2y) m

A B

D

C

J

K

L

M

If the diagonals bisect each other, then RSTU will

be a parallelogram, so assume RV = TV and SV = UV.

16y − 10 = 12y + 50

4y − 10 = 50

4y = 60

y = 15

5x + 2y = 2x + 4y

5x + 2 (15) = 2x + 4 (15)

5x + 30 = 2x + 60

3x + 30 = 60

3x = 30

x = 10

UV = 12y + 50 = 12 (15) + 50 = 230 m

US = 2UV = 2 (230) = 460 m

RV = 5x + 2y = 5 (10) + 2 (15) = 80 m

RT = 2RV = 2 (80) = 160 m

The lengths of the paths are 230 m and 160 m.

Possible answer: When you draw ― AC , you form △DAC and △BAC.

― LM is a

midsegment of △DAC, so ― LM ∥

― AC and LM = 1 _ 2

AC. ― JK is a midsegment of

△BAC , so ― JK ∥

― AC and JK = 1 _ 2

AC. This shows that ― LM ∥

― JK and LM = JK,

so ― LM ≅

― JK . A pair of opposite sides of JKLM are both parallel and

congruent, so JKLM is a parallelogram.




JOURNALHave students describe how to use the measures of the sides of a quadrilateral to determine whether it is a parallelogram.

1215 Lesson 24 . 2


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Lesson Performance TaskIn this lesson you’ve learned three theorems for confirming that a figure is a parallelogram.

• If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

• If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

• If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

For each of the following situations, choose one of the three theorems and use it in your explanation. You should choose a different theorem for each explanation.

a. You’re an amateur astronomer, and one night you see what appears to be a parallelogram in the constellation of Lyra. Explain how you could verify that the figure is a parallelogram.

b. You have a frame shop and you want to make an interesting frame for an advertisement for your store. You decide that you’d like the frame to be a parallelogram but not a rectangle. Explain how you could construct the frame.

c. You’re using a toolbox with cantilever shelves like the one shown here. Explain how you can confirm that the brackets that attach the shelves to the box form a parallelogram ABCD.

A D

CB

a. Possible answer: You could take a photograph of the figure and with a protractor measure to see if the opposite angles are congruent. (If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.)

b. Possible answer: Choose two lengths of framing materials of one length and two of another. (If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.) As long as the angles of the frame are not right angles, the frame is not a rectangle.

c. Possible answer: Attach diagonal braces to opposite corners and check to see if they bisect each other. (If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.)


DO NOT EDIT--Changes must be made through “File info”CorrectionKey=NL-A;CA-A


EXTENSION ACTIVITY

The night sky is divided into 88 constellations, groups of stars that appear to form figures or shapes. Many of the constellations are wholly or partially geometrical in design. Have students research the constellations, looking for triangles and quadrilaterals in the shapes. Students may wish to start with the parallelogram in Lyra, shown in the photo in the Lesson Performance Task Preview. Students may draw or print the shapes they find, and add to them interesting information about the shapes, such as the names of the stars that form them, the distances of the stars, and traditional stories told about the constellations in which the shapes appear.

CONNECT VOCABULARY • What is the converse of a statement? The

converse is the statement that results from

interchanging the hypothesis and conclusion of

the given statement.

• What is the converse of this statement: “If a quadrilateral is a parallelogram, then both pairs of opposite sides are congruent”? If both pairs of

opposite sides of a quadrilateral are congruent,

then the quadrilateral is a parallelogram.

INTEGRATE MATHEMATICAL PRACTICESFocus on Critical ThinkingMP.3 Give an example of a theorem involving quadrilaterals that is true but whose converse is false. Explain why the converse is false. Sample answer:

Theorem: If a figure is a square, then it has four

sides. Converse: If a figure has four sides, then it is a

square. The converse is false because there are

many four-sided figures that are not squares,

including trapezoids and rhombuses that contain

no right angles.

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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