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

CD

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 – If has 1 rt. , then it has 4 rt. s.

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

Given:

Prove:

Example 1:

1. 1. Given

Proof:

ReasonsStatements

4. Transitive Property4.

2. Given2.

3. Opposite sides of a parallelogram are .

3.

Example 1:

Given:

Prove:

Prove that if and are the diagonals of , and

Your Turn:

Proof:

ReasonsStatements

1. Given1.

4. Angle-Side-Angle4.

2. Opposite sides of a parallelogram are congruent.

2.

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

3.

Your Turn:

If lines are cut by a transversal, alt. int.

Definition of congruent angles

Substitution

RSTU is a parallelogram. Find and y.

Example 2:

Angle Addition Theorem

Substitution

Subtract 58 from each side.

Example 2:

Substitution

Divide each side by 3.

Definition of congruent segments

Answer:

Example 2:

ABCD is a parallelogram.

Answer:

Your Turn:

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.

Read the Test ItemSince the diagonals of a parallelogram bisect each other, the intersection point is the midpoint of

A B C D

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)?

Example 3:

Solve the Test Item

Find the midpoint of

The coordinates of the intersection of the diagonals of parallelogram MNPR are (1, 2).

Answer: C

Midpoint Formula

Example 3:

Answer: B

A B C D

MULTIPLE-CHOICE TEST ITEM What are the coordinates of the intersection of the diagonals of parallelogram LMNO, with verticesL(0, –3), M(–2, 1), N(1, 5), O(3, 1)?

Your Turn:

Assignment

Pre-AP Geometry:Pg. 414 #13, 14, 16 – 33, 36,

50

Geometry:Pg. 414 #4 – 12, 16 – 31