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Objectives. State the conditions under which you can prove a quadrilateral is a parallelogram. Converse of Theorem 6-1. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Converse of Theorem 6-2.

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Objectives
Objectives

  • State the conditions under which you can prove a quadrilateral is a parallelogram


Converse of theorem 6 1
Converse of Theorem 6-1

If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.


Converse of theorem 6 2
Converse of Theorem 6-2

If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.


Converse of theorem 6 3
Converse of Theorem 6-3

If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.



Theorem 6 8
Theorem 6-8

If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.


Example 1
Example 1

If the diagonals bisect each other, then the quadrilateral is a parallelogram.

2y – 7 = y + 2

y – 7 = 2

y = 9

3x = y

3x = 9

x = 3

For value of x will quadrilateral MNPL be a parallelogram?


Example 2a
Example 2a

Angles A and C are congruent. ∠ADC and ∠CBA are congruent by the Angle Addition Postulate. Since both pairs of opposite angles are congruent, ABCD is a parallelogram.


Example 2b
Example 2b

This cannot be proven because there is not enough information given. It is not stated that the single-marked sides are congruent to the double-marked sides. If opposite sides are congruent, then the quadrilateral is a parallelogram.


Quick check 2a
Quick Check 2a

Since one pair of opposite sides are both parallel and congruent, we can use Theorem 6-8 to prove PQRS is a parallelogram.


Quick check 2b
Quick Check 2b

Not enough information is given. It is not stated that the single-marked segments are congruent to the double-marked segments. If diagonals bisect each other, then the quadrilateral is a parallelogram.