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Ways to Prove that Quadrilaterals are Parallelograms Lesson 5.2 Geometry Honors Page 172 Lesson Focus There are five ways to prove that a quadrilateral is a parallelogram. Four theorems are presented in this lesson that can be used to show that a quadrilateral is a parallelogram. Theorem 5.4 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 5.5 If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. Theorem 5.6 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Theorem 5-7 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. Examples Complete with always, sometimes, or never. 1. The diagonals of a quadrilateral _______ bisect each other. 2. If the measures of two angles of a quadrilateral are equal, then the quadrilateral is _______ a parallelogram. 3. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is _______ a parallelogram. 4. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is _______ a parallelogram. 5. To prove a quadrilateral is a parallelogram, it is _______ enough to show that one pair on opposite sides is parallel. Examples Complete with always, sometimes, or never. 1. The diagonals of a quadrilateral sometimes bisect each other. 2. If the measures of two angles of a quadrilateral are equal, then the quadrilateral is sometimes a parallelogram. 3. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is always a parallelogram. 4. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is always a parallelogram. 5. To prove a quadrilateral is a parallelogram, it is never enough to show that one pair on opposite sides is parallel. Practice State the principal definition or theorem that enables you to deduce, from the information given, that quad. ABCD is a parallelogram. B C 1. BE = ED; CE = EA 2. BAD DCB; ADC CBA 3. BC AD; AB DC 4. BC AD; AB DC 5. BC AD; BC AD E A D Homework Page 174 - 176 Problems 9, 11, 13, 15, 17, 18, 21, 24