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 = (9-2)(180) 9 = 140 2x + 3 = 11 2x = 8 x = 4

Homework Answers: Parallelograms 5 Characteristics of a parallelogram 1. Both pairs of opposite edges are parallel 2. Both pairs of opposite edges are congruent 3. Both pairs of opposite angles are congruent 4. Diagonals bisect each other 5. Consecutive angles are supplementary 6. x = 78. x = 7 7. x = 209. x = 5

10. x = 3, y = x = 4, y = YES 18. YES19. YES StatementReason 1.ABCD is a parallelogram 1. given Both pairs of opp. sides of a para. are Alternate interior angles are congruent reflexive ASA

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

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° 55° 90°

Get in your …. 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 congruent The other pair are parallel by alt int ∠ ≅ Not a Parallelogram Both pairs of opposite sides are parallel Parallelogram One pair of opposite sides are congruent The same pair are parallel by alt int ∠ ≅ Parallelogram

One pair of opposite sides are congruent The 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 congruent The other pair are parallel Not a Parallelogram

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 Proving that a Quadrilateral is a Parallelogram

Is this a parallelogram?______ justify your conclusion. 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 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

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. 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 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 YES

Answer Essential Question #2 How can we prove a quadrilateral is a parallelogram? Get in your …. To….