1. Recall Theorem 6-3 The diagonals of a parallelogram bisect each other.
Section 6-3 Prove that Quadrilateral is a Parallelogram SPI 32 H: apply properties of quadrilaterals to solve a real-world problem
Objectives:
Determine whether a quadrilateral is a parallelogram
Recall Theorem 6-3 The diagonals of a parallelogram bisect each other.

2. If a diagram shows two diagonals bisect then it is a parallelogram.
Theorem 6-5 Converse of Theorem 6-3 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.
If a diagram shows two diagonals bisect
then it is a parallelogram.

3. Theorem 6-6 If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram.

4. Use Theorems to Solve Problems
Find values of x and y for which ABCD must be a parallelogram.
If the diagonals of quadrilateral ABCD bisect each other, then ABCD is a parallelogram by Theorem 6-5. Write and solve two equations to find values of x and y for which the diagonals bisect each other.
10x – 24 = 8x Diagonals of parallelograms 2y – 80 = y + 9
bisect each other.
2x – 24 = y – 80 = 9
Collect the variable terms
on one side.
x = 18
2x = 36
y = 89
Solve.
If x = 18 and y = 89, then ABCD is a parallelogram.

5. Theorem 6-7 If both pairs of opposite sides of a quadrilateral are congruent, the quadrilateral is a parallelogram.

6. Theorem 6-8 If both pairs of opposite angles of a quadrilateral are congruent, the quadrilateral is a parallelogram.

7. Use Theorems to Solve Problems
Determine whether the quadrilateral is a parallelogram. Explain. a.
a. All you know about the quadrilateral is that only one pair of opposite sides is congruent.
Therefore, you cannot conclude that the quadrilateral is a parallelogram. b.
b. The sum of the measures of the angles of a polygon is (n – 2)180, where n represents the number of sides, so the sum of the measures of the angles of a quadrilateral is (4 – 2)180 = 360.
If x represents the measure of the unmarked angle, x = 360, so x = 105.
Theorem 6-8 states If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Because both pairs of opposite angles are congruent, the quadrilateral is a parallelogram by Theorem 6-8.

8. Parallelograms and Real-world
Dallas Center
Waterfront Park in Vancouver B.C.