Assignment

• P. 518-521: 1, 2, 4-16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48

• P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36

• Challenge Problems

Parallelograms

What makes a polygon a parallelogram?

8.2 Use Properties of Parallelograms

Objectives:

1. To discover and use properties of parallelograms

2. To find side, angle, and diagonal measures of parallelograms

3. To find the area of parallelograms

The Story of Parallelograms

For this lesson, we’ll be writing The Story of Parallelograms. While short on plot, the story is definitive and is illustrated. In order to write this book, we’ll need to make a small, 8-page booklet with no staples. That part is magic.

The Story of Parallelograms: Page Layout

1. Title 2. Definition 3. Picture 4. Theorem 1

5. Theorem 2 6. Theorem 3 7. Theorem 4 8. Area

Parallelogram

A parallelogram is a quadrilateral with both pairs of opposite sides parallel.

• Written PQRS • PQ||RS and QR||PS

Investigation 1

In this Investigation, we will be using Geometer’s Sketchpad to construct a perfect parallelogram, and then we will discover four useful properties about parallelograms.

Theorem 1

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

If PQRS is a parallelogram, then and . RSPQ PSQR

Theorem 2

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

If PQRS is a parallelogram, then and .

RP SQ

Theorem 3

If a quadrilateral is a parallelogram, then consecutive angles are supplementary.

If PQRS is a parallelogram, then x + y = 180°.

Theorem 4

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

Example 1

Find each indicated measure.

1. NM

2. KM

3. mJKL4. mLKM

Example 2

The diagonals of parallelogram LMNO intersect at point P. What are the coordinates of P?

Example 3

Find the values of c and d.

Example 4

For the parallelogram below, find the values of t and v.

Example 5: SAT

For parallelogram ABCD, if AB > BD, which of the following statements must be true?

I. CD < BD

.II ADB > C.III CBD > A

C

A D

B

Bases and Heights

Any one of the sides of a parallelogram can be considered a base. But the height of a parallelogram is not necessarily the length of a side.

Bases and Heights

The altitude is any segment from one side of the parallelogram perpendicular to a line through the opposite side. The length of the altitude is the height.

Bases and Heights 2

The altitude is any segment from one side of the parallelogram perpendicular to a line through the opposite side. The length of the altitude is the height.

Investigation 2

Now you will discover a formula for computing the area of a parallelogram.

Area of a Parallelogram Theorem

The area of a parallelogram is the product of a base and its corresponding height.

Base (b)

Height (h)

Base (b)

Height (h)

A = bh

Area of a Parallelogram Theorem

The area of a parallelogram is the product of a base and its corresponding height.

A = bh

Example 6

Find the area of parallelogram PQRS.

Example 7

What is the height of a parallelogram that has an area of 7.13 m2 and a base 2.3 m long?

Example 8

Find the area of each triangle or parallelogram.1. 2. 3.

Example 9

Find the area of the parallelogram.

Assignment

• P. 518-521: 1, 2, 4-16 even, 24, 26, 28-33, 39, 40, 42, 43, 46-48

• P. 723-724: 1, 3, 4, 10, 11, 17, 23, 24, 26, 36

• Challenge Problems