8.2

Parallelograms

Objectives

 Recognize and apply properties of the sides and angles of parallelograms.

 Recognize and apply properties of the diagonals of parallelograms.

Parallelograms

 A quadrilateral with parallel opposite sides is called a

parallelogram

( ABCD). A B D C

Parallelograms Theorems



Theorem 8.3

– Opposite sides of are ≅ .



Theorem 8.4

– Opposite  s in are ≅ .



Theorem 8.5

– Consecutive  s in are supplementary.



Theorem 8.6

rt.  s.

– If has 1 rt.  , then it has 4

Example 1:

Prove that if a parallelogram has two consecutive sides congruent, it has four sides congruent.

Given: Prove:

Example 1:

Proof: Statements

1.

2.

3.

4.

Reasons

1. Given 2. Given 3. Opposite sides of a parallelogram are  .

4. Transitive Property

Your Turn:

Prove that if and are the diagonals of , and Given: Prove:

Your Turn:

1.

2.

Proof: Statements

3.

4.

Reasons

1. Given 2. Opposite sides of a parallelogram are congruent.

3. If 2 lines are cut by a transversal, alternate interior s are  .

4. Angle-Side-Angle

Example 2:

RSTU is a parallelogram. Find and y. If lines are cut by a transversal, alt. int.

Definition of congruent angles Substitution

Example 2:

Angle Addition Theorem Substitution Subtract 58 from each side.

Example 2:

Definition of congruent segments Substitution Divide each side by 3.

Answer:

Your Turn:

ABCD is a parallelogram.

Answer:

Diagonals of Parallelograms



Theorem 8.7

– The diagonals of a bisect each other.



Theorem 8.8

– Each diagonal of a separates the into two ≅

∆

s.

Example 3:

MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the

diagonals of parallelogram MNPR, with vertices

M

( –3, 0),

N

( –1, 3),

P

(5, 4), and R (3, 1)?

A B C D Read the Test Item

Since the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of

Example 3:

Solve the Test Item

Find the midpoint of Midpoint Formula The coordinates of the intersection of the diagonals of parallelogram

MNPR

are (1, 2).

Answer:

C

Your Turn:

MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the

diagonals of parallelogram LMNO, with vertices

L

(0, –3),

M

( –2, 1),

N

(1, 5),

O

(3, 1)?

A B C D Answer:

B

Assignment



Pre-AP Geometry:

Pg. 414 #13, 14, 16 – 33, 36, 50 

Geometry:

Pg. 414 #4 – 12, 16 – 31