1. Proving Quadrilaterals are Parallelograms Section 6.3 November 16, 2001

2. Prove a quadrilateral is a parallelogram. Use coordinate geometry with parallelograms. Goals: Today you will learn to

3. Theorems Proving Parallelograms Theorem 6.6: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. A BC D Given: AB  CD and AD  BC, Conclusion: ABCD is a parallelogram

4. Theorems Proving Parallelograms Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Given:  A   C and  B  D Conclusion: ABCD is a parallelogram A B C D ) ) ))

5. Theorems Proving Parallelograms Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. A B C D xº (180-x)º Given:  B is supplementary to  A and  B is supplementary to  C Conclusion: ABCD is a parallelogram

6. Theorems Proving Parallelograms Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Given: AC bisects BD and BD bisects AC Conclusion: ABCD is a parallelogram A B C D 

7. Theorems Proving Parallelograms Theorem 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. Given: AD  BC, AD ll BC Conclusion: ABCD is a parallelogram A BC D 

8. Six Ways to Prove Quadrilaterals are Parallelograms Summary 1.Show both pairs of opposite sides are parallel. 2.Show both pairs of opposite sides are congruent. 3.Show both pairs of opposite angles are congruent. 4.Show both diagonals bisect each other. 5.Show one angle is supplementary to both consecutive angles. 6.Show one pair of opposite sides are both congruent and parallel.

9. Example #1 Given:  PQT   RST Prove: PQRS is a parallelogram T P Q R S

10. Example #2 Given: ABCD is a parallelogram FE ll DC Prove: ABEF is a parallelogram A F D C E B

11. Example #3 Given: HJKM is a parallelogram  IJK  LMH Prove: HIKL is a parallelogram M H I J K L 1 2

12. Example #4 Given:  1  2,  IJK  LMH Prove: HIKL is a parallelogram M H I J K L 1 2

13. Example #5 Show that A(-1,2), B(3,2), C(1,-2) and D(-3,-2) are the vertices of a parallelogram by a.Showing both pair of opposite sides are parallel. b.Showing both pair of opposite sides are congruent.

14. Example #6 Identify any quadrilateral that is a parallelogram. a.G(-3,1), H(4,1), I(3,6), J(-1,6) b.P(-2,2), Q(1,1), R(4,4), S(1,4) c.W(3,-1), X(4,2), Y(1,5), Z(0,2)

15. Homework Pg 342 #1-19, 25, 29, 39-47