1. What is the measure of one exterior angle of a regular octagon?

Warm-up 2/1/12

2. What is the sum of the exterior angles of a heptagon?

3. What is the measure of one interior angle of a regular nonagon?

4. If ABCD is a parallelogram, find the value of x.

2x + 3

11

3608

= 45

360

(9-2)(180)9

= 140

2x + 3 = 112x = 8 x = 4

Homework Answers: Parallelograms

5 Characteristics of a parallelogram1. Both pairs of opposite edges are parallel2. Both pairs of opposite edges are congruent3. Both pairs of opposite angles are congruent4. Diagonals bisect each other5. Consecutive angles are supplementary

6. x = 7 8. x = 77. x = 20 9. x = 5

10. x = 3, y = 611. x = 4 , y = 6

12.

13. 16. 14. 17.15. YES 18. YES 19. YES

Statement Reason

1. ABCD is a parallelogram 1. given2. 2. Both pairs of opp. sides of a para. are 3. 3. Alternate interior angles are congruent 4. 4. reflexive5. 5. ASACDAABC

CBADAC ;CD 43;21

ACAC

3

2m

3

2m

3

2m

3

2m

20. YES

4.3 Five Ways to Prove a Quadrilateral is a Parallelogram.

Essential Questions:(1). What are the characteristics of a parallelogram?

(2). How can we prove a quadrilateral is a parallelogram?

Essential Question #1

What are the characteristics of a parallelogram?

Consider the parallelograms drawn below to answer questions 1 – 5. Measure angles or edges as necessary to make your conclusions:

1. Both pairs of opposite edges are parallel?

2. Both pairs of opposite edges are congruent?

T

T

2.52.5

4.4

4.4

2.1

5.4

5.4

2.1

 3. Both pairs of opposite angles are congruent?

4. Diagonals bisect each other?

5. The same pair of opposite edges are congruent and parallel?

T

T

T

125°

125°

55°

55°

90°

90°

90°

90°

Get in your ….

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Answer Essential Question #1

What are the characteristics of a parallelogram?

To….

Essential Question #2

How can we prove a quadrilateral is a parallelogram?

Now identify the five ways to prove a quadrilateral is a parallelogram and list them below:

1. _______________________________________

2. _______________________________________

3. _______________________________________

4. _______________________________________

5. _______________________________________

Show both pairs of opposite sides are parallel

Show both pairs of opposite sides are congruent

Show both pairs of opposite angles are congruent

Show the diagonals bisect each other

Show same pair of opposites sides are both parallel and congruent

Look at the figures below and the given information that is marked. Can the figure be proved to be a parallelogram using one of the methods listed above?

Both pairs of opposite sides are congruent

Parallelogram

One pair of opposite sides are congruentThe other pair are parallel by alt int ∠ ≅Not a Parallelogram

Both pairs of opposite sides are parallel

Parallelogram

One pair of opposite sides are congruentThe same pair are parallel by alt int ∠ ≅Parallelogram

One pair of opposite sides are congruentThe same pair are parallel

Parallelogram

Diagonals bisect each other

Parallelogram

Both pairs of opposite sides are parallel by alt int ∠≅Parallelogram

One pair of opposite sides are congruentThe other pair are parallel

Not a Parallelogram

212

212 yyxxd

Method 1: Using Distance Formula

Ex 1. The coordinates of a Quadrilateral are W (5,6), X (7,5), Y (9,9) and Z (7,10). Use the distance formula to find each edge length.

a. Find WX

b. Find XY

5146557 22 WX

52201645979 22 XY

Proving that a Quadrilateral is a Parallelogram

Is this a parallelogram?______ justify your conclusion.

51491097 22 YZ

522016410675 22 WZ

Both pairs of opposite sides are congruent

Ex 1 continued - The coordinates of a Quadrilateral are W (5,6), X (7,5), Y (9,9) and Z (7,10). c. Find YZ

d. Find WZ

YES

Method 2: Using the Slope Formula

12

12

xx

yym

Ex 2. The coordinates of the Quadrilateral are W(-3,-3), X(1,-6), Y(5,-3) and Z(1,0). a. Find the slope of WX 

b. Find the slope of XY 

4

3

31

36

WXm

4

3

15

63

XYm

Ex 2. The coordinates of the Quadrilateral are W(-3,-3), X(1,-6), Y(5,-3) and Z(1,0). c. Find the slope of YZ 

d. Find the slope of WZ   Is this a parallelogram?_______ justify your conclusion.

4

3

51

30

YZm

4

3

31

30

WZm

Both pairs of opposite edges are Parallel.

YES

Classwork: Use which ever method you chose to determine if a quadrilateral with the following vertices is a Parallelogram. 1. A ( 3, -2) B ( 8, -2) C ( 7, -4) D ( 3, -6)

    

IS ABCD a Parallelogram? _______ If so , How ?

No

525025)2(238 22 AB

541)2(487 22 BC

20416)4(673 22 CD

416160)2(633 22 AD

Opposite Sides are not Congruent.

2. A ( -1, 3) B ( 2, 1) C ( 9, 2) D ( 6, 4)

IS ABCD a Parallelogram? ______ If so , How ?

Both pairs of opposite sides are parallel

3

2

)1(2

31

ABm7

1

29

12

BCm

3

2

96

24

CDm 7

1

)1(6

34

ADm

YES

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Answer Essential Question #2

How can we prove a quadrilateral is a parallelogram?

Get in your ….

To….