1. Proving Quadrilaterals Are Parallelograms
Chapter 6 Section 6.3
Proving Quadrilaterals Are Parallelograms

2. If a quadrilateral is a parallelogram, then …
Quick Review
Some Properties of Parallelograms
Definition: A parallelogram is a quadrilateral with both pairs of opposite sides congruent
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

3. Quadrilateral PQRS with both pairs of opposite sides congruent
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

4. Quadrilateral PQRS with both pairs of opposite angles congruent
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

5. PQRS is a parallelogram
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

6. PQRS is a parallelogram
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
and
PQRS is a parallelogram

7. PQRS is a parallelogram
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
and
PQRS is a parallelogram

8. Yes, one pair of opposite sides congruent and parallel
Proving a Quad is a Parallelogram
Yes, one pair of opposite sides congruent and parallel
Yes, diagonals bisect each other

9. Yes, both pairs of opposite sides congruent
Proving a Quad is a Parallelogram
Yes, both pairs of opposite sides congruent
No, same pair of opposite sides must be parallel and congruent

10. Yes, could show both pairs of opposite sides are parallel
Proving a Quad is a Parallelogram
Yes, could show both pairs of opposite sides are parallel
Yes, Consecutive angles are supplementary

11. Definition: Theorem 6.10: Theorem 6.6: Theorem 6.10: Definition:
Proving a Quad is a Parallelogram
Definition:
Theorem 6.10:
Theorem 6.6:
Theorem 6.10:
Definition:
Theorem 6.10:

12. Theorem 6.9: Theorem 6.8: mCDA + mDCB = 180 OR mDAB + mABC = 180
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 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 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 = y
110 = y
75 = 3y
25 = y

15. 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 a Quad is a Parallelogram
2. Def.  ’s

17. Proving a Quad is a Parallelogram
2. Def.  ’s

18. HW #68 Pg Even, 30, 32, 34-37 