6.3 proving quadrilaterals are parallelograms · how to prove that a quadrilateral is a ... ex. 4:...

6.3 Proving

Quadrilaterals are

Parallelograms

Geometry

Mr. Peebles

Spring 2012

Bell Ringer –

PQRS is a parallelogram.

If mP = 83 Degrees, then

Find the angle measure.

a. mR

b. mQ

P

R Q

83° S

Objectives:

Daily L:earning Target (DLT)

Friday November 30, 2012 “I can understand, apply, and remember

how to prove that a quadrilateral is a

parallelogram.”

Bell Ringer –




a. mR = 83

b. mQ = 97

P

R Q

83° S

Assignment

pp. 324-327 #1-9, 18, 29-30

Assignment 1. x = 5 9. It’s a

parallelogram

2. x = 3, y = 4 18. D

3. x = 1.6, y = 1 29. C

4. x = 5/3 30. F

5. x = 5

6. x = 13

7. It’s a parallelogram

8. It’s NOT a parallelogram

Theorems

Theorem 6.6: If both

pairs of opposite

sides of a

quadrilateral are

congruent, then the

quadrilateral is a

parallelogram.

A

D

B

C

ABCD is a parallelogram.

Theorems

Theorem 6.7: If both

pairs of opposite

angles of a

quadrilateral are

congruent, then the

quadrilateral is a

parallelogram.

A

D

B

C


Theorems

Theorem 6.8: If an

angle of a

quadrilateral is

supplementary

to both of its

consecutive

angles, then the

quadrilateral is a

parallelogram.

A

D

B

C


x°

(180 – x)° x°

Theorems

Theorem 6.9: If the

diagonals of a

quadrilateral

bisect each other,

then the

quadrilateral is a

parallelogram. ABCD is a parallelogram.

A

D

B

C

Ex. 2: Proving Quadrilaterals are

Parallelograms

As the sewing box below is opened, the

trays are always parallel to each other.

Why?

2.75 in. 2.75 in.

2 in.

2 in.

Ex. 2: Proving Quadrilaterals are

Parallelograms

Each pair of hinges are opposite sides of a quadrilateral. The 2.75 inch sides of the quadrilateral are opposite and congruent. The 2 inch sides are also opposite and congruent. Because opposite sides of the quadrilateral are congruent, it is a parallelogram. By the definition of a parallelogram, opposite sides are parallel, so the trays of the sewing box are always parallel.

2.75 in. 2.75 in.

2 in.

2 in.

Another Theorem ~

Theorem 6.10—If one pair of opposite

sides of a quadrilateral are congruent and

parallel, then the quadrilateral is a

parallelogram.

ABCD is a

parallelogram.

A

B C

D

Objective 2: Using Coordinate Geometry

When a figure is in the coordinate plane,

you can use the Distance Formula (see—it

never goes away) to prove that sides are

congruent and you can use the slope

formula (see how you use this again?) to

prove sides are parallel.

Ex. 4: Using properties of parallelograms

Show that A(2, -1), B(1,

3), C(6, 5) and D(7,1)

are the vertices of a

parallelogram.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Ex. 4: Using properties of parallelograms Method 1—Show that opposite

sides have the same slope, so they are parallel.

Slope of AB. 3-(-1) = - 4

1 - 2

Slope of CD. 1 – 5 = - 4

7 – 6

Slope of BC. 5 – 3 = 2

6 - 1 5

Slope of DA. - 1 – 1 = 2

2 - 7 5

AB and CD have the same slope, so they are parallel. Similarly, BC ║ DA.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Because opposite sides are

parallel, ABCD is a

parallelogram.


Method 2—Show that

opposite sides have the

same length.

AB=√(1 – 2)2 + [3 – (- 1)2] = √17

CD=√(7 – 6)2 + (1 - 5)2 = √17

BC=√(6 – 1)2 + (5 - 3)2 = √29

DA= √(2 – 7)2 + (-1 - 1)2 = √29

AB ≅ CD and BC ≅ DA.

Because both pairs of opposites

sides are congruent, ABCD is a

parallelogram.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)


Method 3—Show that

one pair of opposite

sides is congruent and

parallel.

Slope of AB = Slope of CD

= -4

AB=CD = √17

AB and CD are congruent

and parallel, so ABCD is a

parallelogram.

6

4

2

-2

-4

5 10 15

D(7, 1)

C(6, 5)

B(1, 3)

A(2, -1)

Quadrilateral Family Tree Construction Project -

Each Group Shall Create The Following

Quadrilaterals:

1. Quadrilateral any size you want

2. 5 x 2 Parallelogram

3. 6 x 2 Rectangle

4. 4 x 4 Square

5. 3 x 3 Rhombus

6. 3 x 4 Kite

7. Isosceles Trapezoid any size you want

8. 8 arrows needed for the Quad Tree

Quadrilateral Family Tree Construction Project

Rubric –

5 Points Each

1. Quadrilaterals Measure Correctly

2. Quadrilaterals Are Cut With Lines

Shown

3. Quadrilaterals Are Appropriately

Labeled

4. At Least 8 arrows are needed for your

Quadrilateral Tree Project

5. Projects Are Neat

Closer – Exit Quiz – 4 Points




a. mR

b. mQ

P

R Q

81° S

6.3 proving quadrilaterals are parallelograms · how to prove that a quadrilateral is a ... ex. 4:...

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