Name Class Date

H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

Explore Investigating ParallelogramsA 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.

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.

Identify the opposite sides and opposite angles of the parallelogram.

Opposite sides:

Opposite angles:

Resource Locker

Side _ AB is opposite side

_ DC. Side

_ AD is opposite side

_ BC.

A is opposite C. B is opposite D.

Module 24 1189 Lesson 1

24 . 1 Properties of ParallelogramsEssential Question: What can you conclude about the sides, angles, and diagonals

of a parallelogram?

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

IN1_MNLESE389762_U9M24L1.indd 1189 4/18/14 7:47 PM

Common Core Math StandardsThe student is expected to:

G-CO.11

Prove theorems about parallelograms. Also G-SRT.5

Mathematical Practices

MP.3 Logic

Language ObjectiveExplain to a partner why pictures of quadrilaterals are or are not parallelograms.

HARDCOVER PAGES 965974

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

Properties of Parallelograms

ENGAGE Essential Question: What can you conclude about the sides, angles, and diagonals of a parallelogram?Opposite sides are congruent, opposite angles are

congruent, consecutive angles are supplementary,

and diagonals bisect each other.

PREVIEW: LESSON PERFORMANCE TASKView 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.

1189

HARDCOVER

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

Name

Class Date

H

ough

ton

Mif

flin

Har

cour

t Pub

lishi

ng C

omp

any

Explore Inves

tigating Parallelogr

ams

A quadrilateral is a pol

ygon with four sides. A p

arallelogram is a quadril

ateral that

has two pairs of parallel

sides. You can use geom

etry software to investig

ate properties

of parallelograms.

Draw a stra

ight line. Then plot a po

int that is not on the lin

e. Construct a line

through the point that i

s parallel to the line. Th

is gives you a pair of pa

rallel lines.

Repeat Step

A to construct a second

pair of parallel lines tha

t intersect those from

Step A.

The intersec

tions of the parallel line

s create a parallelogram

. Plot

points at these intersecti

ons. Label the points A

, B, C, and D.

Identify the oppos

ite sides and opposite an

gles of the parallelogram

.

Opposite sides:

Opposite angles:

Resource

Locker

G-CO.11 Prove theorem

s about parallelogram

s. Also G-SRT.5

Side _ AB is oppos

ite side _ DC. Side

_ AD is opposite side

_ BC.

A is opposite C. B i

s opposite D.

Module 24

1189

Lesson 1

24 . 1 Properties of Pa

rallelograms

Essential Question

: What can you conc

lude about the sides

, angles, and diagon

als

of a parallelogram?

DO NOT EDIT--Chan

ges must be made th

rough "File info"

CorrectionKey=NL-A

;CA-A

IN1_MNLESE389762_U

9M24L1.indd 1189

4/18/14 7:44 PM

1189 Lesson 24 . 1

L E S S O N 24 . 1

DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-C;CA-C

H

oug

hton Mifflin H

arcourt Publishin

g Com

pany

Measure each angle of the parallelogram.

Measure the length of each side of the parallelogram. You can do this by measuring the distance between consecutive vertices.

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.

Conjecture:

A segment that connects two nonconsecutive vertices of a polygon is a diagonal. Construct diagonals AC and

_ BD . Plot a point at the intersection of the diagonals and

label it E.

Measure the length of AE , BE , CE , and DE .

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.

Conjecture:

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:

Opposite sides of a parallelogram are congruent. Opposite

angles of a parallelogram are congruent.

The diagonals of a parallelogram bisect each other.

Consecutive angles of a parallelogram are supplementary.

Module 24 1190 Lesson 1

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

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

IN1_MNLESE389762_U9M24L1.indd 1190 4/18/14 7:47 PM

Math BackgroundIn 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 parallelograms 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.

EXPLORE Investigating Parallelograms

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

QUESTIONING STRATEGIESAs you drag points, does the quadrilateral remain a parallelogram? 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.

PROFESSIONAL DEVELOPMENT

Properties of Parallelograms 1190

DO NOT EDIT--Changes must be made through File infoCorrectionKey=NL-C;CA-C

H

oug

hton

Mif

flin

Har

cour

t Pub

lishi

ng

Com

pan

y

2. 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 students statement.

Explain 1 Proving Opposite Sides Are CongruentThe 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.

Example 1 Prove that the opposite sides of a parallelogram are congruent.

Given: ABCD is a parallelogram.

Prove: _ AB _ CD and

_ AD _ CB

Statements Reasons

1. ABCD is a parallelogram. 1.

2. Draw _ DB . 2. Through any two points, there is exactly

one line.

3. _ AB _ DC , _ AD _ BC 3.

4. ADB CBDABD CDB 4.

5. _ DB

_ DB 5.

6. 6. ASA Triangle Congruence Theorem

7. _ AB

_ CD and

_ AD

_ CB 7.

Reflect

3. Explain how you can use the rotational symmetry of a parallelogram to give an argument that supports the above theorem.

A B

CD

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.

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.

Given

Definition of parallelogram

Alternate Interior Angles Theorem

Reflexive Property of Congruence

CPCTC

ABD CDB

Module 24 1191 Lesson 1

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

IN1_MNLESE389762_U9M24L1 1191 9/30/14 12:02 PM

COLLABORATIVE LEARNING