6.2 Properties of Parallelograms6.2 Properties of Parallelograms

Geometry

Mrs. Spitz

Spring 2005

Objectives:Objectives:

• Use some properties of parallelograms.

• Use properties of parallelograms in real-lie situations such as the drafting table shown in example 6.

Assignment:Assignment:

• pp. 333-335 #2-37 and 39

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

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

►PQ RS and ≅SP QR≅ P

Q R

S

Theorems about parallelogramsTheorems about parallelograms

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

P ≅ R andQ ≅ S

P

Q R

S

Theorems about parallelogramsTheorems about parallelograms

• 6.4—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

Theorems about parallelogramsTheorems about parallelograms

• 6.5—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 Ex. 1: Using properties of ParallelogramsParallelograms• 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 Ex. 1: Using properties of ParallelogramsParallelograms• FGHJ is a parallelogram.

Find the unknown length. Explain your reasoning.

a. JHb. JK

SOLUTION: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 Ex. 1: Using properties of ParallelogramsParallelograms• FGHJ is a parallelogram.

Find the unknown length. Explain your reasoning.

a. JHb. JK

SOLUTION:a. JH = FG Opposite sides

of a are ≅. JH = 5 Substitute 5

for FG.

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. mQ P

RQ

70°

S

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

PQRS is a parallelogram.

Find the angle measure.

a. mR

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

mR = 70° Substitute 70° for mP.

P

RQ

70°

S

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

PQRS is a parallelogram.

Find the angle measure.

a. mR

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

mR = 70° Substitute 70° for mP.

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

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 = 60

x = 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

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

2. Opposite s of a ▭ are ≅

3

2

1

C

D

A

G

BE

F

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

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

congruence.

3

2

1

C

D

A

G

BE

F

Ex. 5: Proving Theorem 6.2Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .2. Draw BD.3. AB ║CD, AD ║ CB.4. ABD ≅ CDB, ADB ≅

CBD5. DB ≅ DB6. ∆ADB ≅ ∆CBD7. AB ≅ CD, AD ≅ CB

1. Given

A

D

B

C

Ex. 5: Proving Theorem 6.2Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .2. Draw BD.

3. AB ║CD, AD ║ CB.4. ABD ≅ CDB, ADB ≅

CBD5. DB ≅ DB6. ∆ADB ≅ ∆CBD7. AB ≅ CD, AD ≅ CB

1. Given2. Through any two points, there

exists exactly one line.

A

D

B

C

Ex. 5: Proving Theorem 6.2Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .2. Draw BD.

3. AB ║CD, AD ║ CB.4. ABD ≅ CDB, ADB ≅

CBD5. DB ≅ DB6. ∆ADB ≅ ∆CBD7. AB ≅ CD, AD ≅ CB

1. Given2. Through any two points, there

exists exactly one line.3. Definition of a parallelogram

A

D

B

C

Ex. 5: Proving Theorem 6.2Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .2. Draw BD.

3. AB ║CD, AD ║ CB.4. ABD ≅ CDB, ADB ≅

CBD5. DB ≅ DB6. ∆ADB ≅ ∆CBD7. AB ≅ CD, AD ≅ CB

1. Given2. Through any two points, there

exists exactly one line.3. Definition of a parallelogram4. Alternate Interior s Thm.

A

D

B

C

Ex. 5: Proving Theorem 6.2Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .2. Draw BD.

3. AB ║CD, AD ║ CB.4. ABD ≅ CDB, ADB ≅

CBD5. DB ≅ DB6. ∆ADB ≅ ∆CBD7. AB ≅ CD, AD ≅ CB

1. Given2. Through any two points, there

exists exactly one line.3. Definition of a parallelogram4. Alternate Interior s Thm.

5. Reflexive property of congruence

A

D

B

C

Ex. 5: Proving Theorem 6.2Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .2. Draw BD.

3. AB ║CD, AD ║ CB.4. ABD ≅ CDB, ADB ≅

CBD5. DB ≅ DB6. ∆ADB ≅ ∆CBD7. AB ≅ CD, AD ≅ CB

1. Given2. Through any two points, there

exists exactly one line.3. Definition of a parallelogram4. Alternate Interior s Thm.

5. Reflexive property of congruence6. ASA Congruence Postulate

A

D

B

C

Ex. 5: Proving Theorem 6.2Ex. 5: Proving Theorem 6.2

Given: ABCD is a parallelogram. Prove AB ≅ CD, AD ≅ CB.

1. ABCD is a .2. Draw BD.

3. AB ║CD, AD ║ CB.4. ABD ≅ CDB, ADB ≅

CBD5. DB ≅ DB6. ∆ADB ≅ ∆CBD7. AB ≅ CD, AD ≅ CB

1. Given2. Through any two points, there

exists exactly one line.3. Definition of a parallelogram4. Alternate Interior s Thm.

5. Reflexive property of congruence6. ASA Congruence Postulate7. CPCTC

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

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?

ANSWER: NO. If ABCD were a parallelogram, then by Theorem 6.5, AC would bisect BD and BD would bisect AC. They do not, so it cannot be a parallelogram.

B

C

DA