Properties of ParallelogramsProperties of Parallelograms

Unit 12, Day 1

From the presentation by

Mrs. Spitz, Spring 2005

http://www.taosschools.org/ths/Departments/MathDept/spitz/Courses/GeometryPPTs/6.2%20Parallelograms.ppt

You will need:You will need:

• Index card

• Scissors

• 1 piece of tape

• Ruler

• Protractor

ExplorationExploration

1. Mark a point somewhere along the bottom edge of your paper.

2. Draw a line from that point to the top right corner of the rectangle to form a triangle.

Amy King

ExplorationExploration

3. Cut along this line to remove the triangle.

4. Attach the triangle to the left side of the rectangle.

5. What shape have you created?

Amy King

In this lesson . . . In this lesson . . .

And the rest of the unit, 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 above, PQ║RS and QR║SP. The symbol PQRS is read “parallelogram PQRS.”

Q R

SP

ExplorationExploration

Measure the lengths of the sides of your parallelogram.

What conjecture could you make regarding the lengths of the sides of a parallelogram?

Amy King

Theorems about parallelogramsTheorems about parallelograms

• 9-1—If a quadrilateral is a parallelogram, then its opposite sides are congruent.

►PQ RS and ≅SP QR≅ P

Q R

S

ExplorationExploration

Measure the angles of your parallelogram.

What conjecture could you make regarding the angles of a parallelogram?

Amy King

Theorems about parallelogramsTheorems about parallelograms

• 9-2—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

• 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

ExplorationExploration

Draw both of the diagonals of your parallelogram.

Measure the distance from each corner to the point where the diagonals intersect.

Amy King

ExplorationExploration

What conjecture could you make regarding the lengths of the diagonals of a parallelogram?

Amy King

Theorems about parallelogramsTheorems about parallelograms

• 9-3—If a quadrilateral is a parallelogram, then its diagonals bisect each other.

QM ≅ SM and PM ≅ RM

P

Q R

S

M

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

parallelogram. Find the unknown length.

a. JH

b. JK

F G

J H

K

5

3

b.

Ex. 1: Using properties of Ex. 1: Using properties of ParallelogramsParallelograms

SOLUTION:

a. JH = FG so JH = 5

F G

J H

K

5

3

b.

Ex. 1: Using properties of Ex. 1: Using properties of ParallelogramsParallelograms

SOLUTION:

b. JK = GK, so JK = 3

F G

J H

K

5

3

b.

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

a. mR = mP , so mR = 70°

P

RQ

70°

S

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

b. mQ + mP = 180°

mQ + 70° = 180°

mQ = 110°

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

S

QP

R3x° 120°

Ex. 4: Using parallelograms in real lifeEx. 4: 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. 4: Using parallelograms in real lifeEx. 4: 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

HomeworkHomework

Work Packets: Properties of Parallelograms