Properties of Parallelograms

Parallelogram• A quadrilateral with both pairs of opposite sides parallel

Properties

Opposite sides are congruent

Opposite angles are congruent

Consecutive angles are supplementary.

ADD UP TO 180°

Diagonals bisect each other.

This cuts the diagonals into two equal parts.

• A diagonal makes 2 congruent triangles

ABCD is a parallelogram. Find the lengths and the angle measures.

1. AD

2. EC

3. mADC

4. mBCD

B 8 C

A D

6545

E

5

85

110

70

Find the value of each variable in the parallelogram.

4y

x = 5

2y42x – 6

y = 30

Proofs Involving Parallelograms A E B 2

1D C

3G F

Given: ABCD and AEFG are parallelograms

Prove: <1 = < 3

Statements Reasons

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Plan: Show that both angles are congruent to <2

1. ABCD & AEFG are Parallelograms 1. Given2. <1 = < 2 2. Opposite Angles3. <2 = <3 3. Opposite Angles 4. <1 = <3 4. Transitive ~

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Proving Theorem 6.2

Given: ABCD is a parallelogram

Prove: AB = CD, AD = CB

Statements Reasons

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A B

D C

Plan: Insert a diagonal which will allow us to divide the parallelogram into two triangles

1. ABCD is a parallelogram 1. Given2. AB || CD, and AD || CB 2. Def. of a parallelogram

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3. <ABD = < CDB 3. Alternate Int4. <ADB = < CBD 4. Alternate Int6. DB = DB 6. Reflexive7. /\ ADB = /\ CBD 7. ASA8. AB = CD, AD = CB 8. CPCTC

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Given: /\ RQP = /\ ONP R is the midpoint of MQProve: MRON is a parallelogram

Statements Reasons

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Q

R P O

M N

1. /\ RQP = /\ ONP 1. Given R is the midpoint of MQ

2. MR = RQ 2. Definition of a midpoint

3. RQ = NO 3. CPCTC

4. MR = NO 4. Transitive

5. <QRP = < NOP 5. CPCTC

6. MQ || NO 6. Alternate Interior

7. MRON is a parallelogram 7. Definition

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