Warm up 6.3 CSYN Page 303

6-3 Proving That a Quadrilateral is a Parallelogram page 303 Objective: To determine whether a quadrilateral is a parallelogram

And why…. To use a parallel rule to plot a ship’s course, as in Example 3

Real World Connection Frank Lloyd Wright, a famous architect, used parallelograms in designs of many houses, such as the Kraus House in Kirkwood, Missouri.

Theorem 6-5 (converse of Th. 6-3) If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. A B C D

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

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

Theorem 6-8 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. A B C D

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. If x = 18 and y = 89, then ABCD is a parallelogram. 10x – 24 = 8x + 12 Diagonals of parallelograms2y – 80 = y + 9 bisect each other. Proving that a Quadrilateral Is a Parallelogram x = 18 2x = 36 y = 89 Solve. 2x – 24 = 12 y – 80 = 9 Collect the variable terms on one side. 6-3

Another one… Solve for x in which ABCD must be a parallelogram 3x 4x - 5 A B C D

You try one… Turn to page 305 and complete check understanding 1 (top of page).

Determine whether the quadrilateral is a parallelogram. Explain. a. b. a.All you know about the quadrilateral is that only one pair of opposite sides is congruent. 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. Therefore, you cannot conclude that the quadrilateral is a parallelogram. If x represents the measure of the unmarked angle, x = 360, so x = 105. Proving that a Quadrilateral Is a Parallelogram 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

You try one… Turn to page 306 and complete check understanding 2 (top of page)

Another one… Determine if the quadrilateral must be a parallelogram. Explain. Yes, both pairs of opp. Sides are congruent

The captain of a fishing boat plots a course toward a school of bluefish. One side of a parallel rule connects the boat with the school of bluefish. The other side makes a 36° angle north of due east on the chart’s compass. Explain how the captain knows in which direction to sail to reach the bluefish. Because both sections of the rulers and the crossbars are congruent, the rulers and crossbars form a parallelogram. Therefore, the angle shown on the chart’s compass is congruent to the angle the boat should travel, which is 36° north of due east. Proving that a Quadrilateral Is a Parallelogram 6-3

Summary State 4 different ways you can show a quadrilateral is a parallelogram. 1. If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram 2. If one pair of opposite sides of a quadrilateral is both congruent and parallel, then the quadrilateral is a parallelogram. 3. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. 4. If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

Assignment 6.3 Page 307 #1-29 (omit 17),