LESSON 6.3 PROVING THAT A QUADRILATERAL IS A PARALLELOGRAM

OBJECTIVE:Determine whether a quadrilateral is a parallelogram

Theorem 6.5If the diagonals of a quadrilateral

_______ each other, then the quadrilateral is a parallelogram.

Theorem 6.6If one pair of opposite sides of a

quadrilateral is both _______ and __________, then the quadrilateral is a parallelogram.

Theorems

bisect

parallel

congruent

Theorem 6.7If both pairs of opposite sides of a

quadrilateral are __________, then the quadrilateral is a parallelogram.

Theorem 6.8If both pairs of opposite angles of a quadrilateral are ___________, then

the quadrilateral is a parallelogram.

Theorems Con’t

congruent

congruent

EXAMPLE #1Find 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 + 92x – 24 = 12

2x = 36

x = 18

y – 80 = 9 y =

89

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 ___________

Yes, b/c both pairs of opposite angles are congruent, which makes it a parallelogram.

____________________________________________________________________________

If x represents the measure of the unknown angle,

x + 75 + 105 + 75 = ______, so x = _______.

(4 – 2)180 = 360

360 105

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