OBJECTIVE: PROVING THAT A QUADRILATERAL IS A PARALLELOGRAM LESSON 6.3 PROVING THAT A QUADRILATERAL IS A PARALLELOGRAM OBJECTIVE: Determine whether a quadrilateral is a parallelogram

Theorems Theorem 6.5 If the diagonals of a quadrilateral _______ each other, then the quadrilateral is a parallelogram. Theorem 6.6 If one pair of opposite sides of a quadrilateral is both _______ and __________, then the quadrilateral is a parallelogram. bisect parallel congruent

Theorems Con’t Theorem 6.7 Theorem 6.8 If both pairs of opposite sides of a quadrilateral are __________, then the quadrilateral is a parallelogram. Theorem 6.8 If both pairs of opposite angles of a quadrilateral are ___________, then the quadrilateral is a parallelogram. congruent congruent

EXAMPLE #1 Find the values for x and y for which ABCD must be a parallelogram. If the diagonals of quadrilateral ABCD _______ each other, then ABCD is a parallelogram, so bisect 10x – 24 = 8x + 12 and 2y – 80 = y + 9 2x – 24 = 12 y – 80 = 9 2x = 36 y = 89 x = 18 If x = 18 and y = 89, then ABCD is a ___________. parallelogram

EXAMPLE #2 Determine whether each quadrilateral is a parallelogram. Explain. a. All you know about the quadrilateral is that only one pair of opposite sides is congruent. Is that enough to prove the quadrilateral is a parallelogram? __________________________________________________________________________________________________________________ No, b/c there is no theorem that states only one pair of opposite sides must be congruent to be a parallelogram.

EXAMPLE #2 Determine whether each quadrilateral is a parallelogram. Explain. b. The sum of the measures of the angles in a polygon is (n – 2)180, so the sum of the measures of the angles of a quadrilateral is ___________ (4 – 2)180 = 360 If x represents the measure of the unknown angle, x + 75 + 105 + 75 = ______, so x = _______. 360 105 ____________________________________________________________________________ Yes, b/c both pairs of opposite angles are congruent, which makes it a parallelogram.

Assignment: pg 307 #1-15, 20-29