2. Proving Quadrilaterals are Parallelograms Lesson 6.3 Chapter 6 Section 6.3 Proving Quadrilaterals Are Parallelograms

3. Proving Quadrilaterals are Parallelograms Lesson 6.3 Some Properties of Parallelograms Quick Review If a quadrilateral is a parallelogram, then … Theorem 6.2 Theorem 6.3 Theorem 6.4 Theorem 6.5 both pairs of opposite sides are congruent both pairs of opposite angles are congruent consecutive angles are supplementary diagonals bisect each other Definition: A parallelogram is a quadrilateral with both pairs of opposite sides congruent

4. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Theorem Theorem 6.6 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram Quadrilateral PQRS with both pairs of opposite sides congruent PQRS is a parallelogram

5. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Theorem Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram Quadrilateral PQRS with both pairs of opposite angles congruent PQRS is a parallelogram

6. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Theorem Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram m  S + m  P = 180 and m  S + m  R = 180 PQRS is a parallelogram

7. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Theorem Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram PQRS is a parallelogram and

8. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Theorem Theorem 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram PQRS is a parallelogram and

9. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Yes, one pair of opposite sides congruent and parallel Yes, diagonals bisect each other

10. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Yes, both pairs of opposite sides congruent No, same pair of opposite sides must be parallel and congruent

11. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Yes, could show both pairs of opposite sides are parallel Yes, Consecutive angles are supplementary

12. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram Theorem 6.9: Theorem 6.8: m  CDA + m  DCB = 180 OR m  DAB + m  ABC = 180

13. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram For a Quadrilateral to be a parallelogram opposite sides must be congruent 3x = 6 x + 2 = y - 1 x = 2 2 + 2 = y – 1 4 = y – 1 5 = y

14. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram For a Quadrilateral to be a parallelogram opposite angles must be congruent 2x = 70 3x + 5 = x + 3y x = 35 3(35) + 5 = 35 + 3y 110 = 35 + 3y 75 = 3y 25 = y

15. Proving Quadrilaterals are Parallelograms Lesson 6.3 Proving a Quad is a Parallelogram For a Quadrilateral to be a parallelogram diagonals must bisect each other 3x = 12 x + y = 5y x = 4 4 + y = 5y 4 = 4y 1 = y

16. Proving Quadrilaterals are Parallelograms Lesson 6.3 Use both slope and distance formula to show one pair of opposite side is congruent and parallel Show that the quadrilateral with vertices A(-1, -2), B(5,3), C(-6,2), and D(0,7) is a parallelogram AB = CDAB // CD

17. Proving Quadrilaterals are Parallelograms Lesson 6.3 HW Pg 342 #’s 9, 10, 12, 17-19 all, & 25-26 Quiz on Thursday!!