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1. Name 6 Different types of Quadrilaterals.

2. Are all Squares considered Rectangles?

3. Are all Parallelograms considered Rectangles?

4. How would you find the third side of a right triangle given two of the sides?

9.1 9.1 Properties of ParallelogramsProperties of Parallelograms

Geometry

Mr. Calise

Objectives:Objectives:

• Use some properties of parallelograms.

• Use properties of parallelograms in real-life situations.

In this lesson . . .In this lesson . . .

And the rest of the chapter, you will study special quadrilaterals. A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

When you mark diagrams of quadrilaterals, use matching arrowheads to indicate which sides are parallel. For example, in the diagram to the right, PQ║RS and QR║SP. The symbol PQRS is read “parallelogram PQRS.”

Theorems about parallelogramsTheorems about parallelograms

• If a quadrilateral is a parallelogram, then its opposite sides are congruent.

► PQ RS and ≅ SP QR≅

P

Q R

S


• If a quadrilateral is a parallelogram, then its opposite angles are congruent.

P ≅ R andQ ≅ S

P

Q R

S

Properties of a ParallelogramProperties of a Parallelogram

1. Opposite Sides are Congruent.

2. Opposite Angles are Congruent.

3. Consecutive Angles are Supplementary.

4. Diagonals Bisect each other.

Parallel Lines Cut By A TransversalParallel Lines Cut By A Transversal

If three or more parallel lines are cut by a transversal and the parts of the transversal are congruent, then the parts of all other transversals are also congruent.

Theorems about parallelogramsTheorems about parallelograms• If a quadrilateral is a

parallelogram, then its consecutive angles are supplementary (add up to 180°).

mP + mQ = 180°,

mQ + mR = 180°,

mR + mS = 180°,

mS + mP = 180° P

Q R

S


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

QM ≅ SM and PM ≅ RM

P

Q R

S

Ex. 1: Using properties of ParallelogramsEx. 1: Using properties of Parallelograms

• FGHJ is a parallelogram. Find the unknown length. Explain your reasoning.

a. JH

b. JK

F G

J H

K

5

3

b.

Ex. 1: Using properties of ParallelogramsEx. 1: Using properties of Parallelograms• FGHJ is a parallelogram.

Find the unknown length. Explain your reasoning.a. JHb. JK

a. JH = FG Opposite sides of a are .≅

JH = 5 Substitute 5 for FG.

F G

J H

K

5

3

b.

Ex. 1: Using properties of ParallelogramsEx. 1: Using properties of Parallelograms

• FGHJ is a parallelogram. Find the unknown length. Explain your reasoning.

a. JH

b. JK

F G

J H

K

5

3

b. b.JK = GK Diagonals of a bisect each other.

JK = 3 Substitute 3 for GK

Ex. 2: Using properties of parallelogramsEx. 2: Using properties of parallelograms

PQRS is a parallelogram.

Find the angle measure.

a. mR

b. mQP

RQ

70°S




a. mR

b. mQ

a. mR = mP Opposite angles of a are ≅.

mR = 70° Substitute 70° for mP.

RQ

70°S




a. mR

b. mQ

b. mQ + mP = 180° Consecutive s of a are supplementary.

mQ + 70° = 180° Substitute 70° for mP.

mQ = 110° Subtract 70° from each side.

P S

RQ

70°S

Ex. 3: Using Algebra with ParallelogramsEx. 3: Using Algebra with Parallelograms

PQRS is a parallelogram. Find the value of x.

mS + mR = 180°

3x + 120 = 180

3x = 60x = 20

Consecutive s of a □ are supplementary.

Substitute 3x for mS and 120 for mR.

Subtract 120 from each side.

Divide each side by 3.

S

QP

R3x°

120°

Ex. 4: Proving Facts about ParallelogramsEx. 4: Proving Facts about Parallelograms

Given: ABCD and AEFG are parallelograms.

Prove 1 ≅ 3.

1. ABCD is a □. AEFG

is a ▭.2. 1 ≅ 2, 2 ≅ 3

3. 1 ≅ 3

1. Given

3

2

1

C

D

A

G

BE

F


1. Given

2. Opposite s of a ▭ are ≅

3

2

1

C

D

A

G

BE

F

• Given: ABCD and AEFG are parallelograms.

• Prove 1 ≅ 3.

1. ABCD is a □. AEFG is a .▭

2. 1 ≅ 2, 2 ≅ 3

3. 1 ≅ 3


1. Given

2. Opposite s of a ▭ are ≅ 3. Transitive prop. of

congruence.

3

2

1

C

D

A

G

BE

F

• Given: ABCD and AEFG are parallelograms.

• Prove 1 ≅ 3.

1. ABCD is a □. AEFG is a .▭

2. 1 ≅ 2, 2 ≅ 3

3. 1 ≅ 3

ProofsProofsGiven: ABCD is a

parallelogram.

Prove AB ≅ CD, AD ≅ CB.

A

D

B

C

Ex. 6: Using parallelograms in real lifeEx. 6: Using parallelograms in real life

FURNITURE DESIGN. A drafting table is made so that the legs can be joined in different ways to change the slope of the drawing surface. In the arrangement below, the legs AC and BD do not bisect each other. Is ABCD a parallelogram?

B

C

DA

ANSWER: NO. If ABCD were a

parallelogram, then by definition of a parallelogram, AC would bisect BD and BD would bisect AC. They do not, so it cannot be a parallelogram.

B

C

DA

HomeworkHomework

• Textbook Page 451• #’s 1 – 10, 14 – 16

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