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Transcript
Proving Quadrilaterals are Parallelograms Standards #4,7,12,17 Section 6.3 Tuesday, November 13, 2012 Goals: Today you will learn to • Prove a quadrilateral is a parallelogram. • Use coordinate geometry with parallelograms. Theorems Proving Parallelograms Theorem 6.6: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Given: B C ABCD and ADBC, Conclusion: A ABCD is a parallelogram D Theorems Proving Parallelograms Theorem 6.7 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. Given: A C and BD Conclusion: ABCD is a parallelogram C B A D Theorems Proving Parallelograms Theorem 6.8 If an angle of a quadrilateral is supplementary to both of its consecutive angles, then the quadrilateral is a parallelogram. Given: B is supplementary to A and B is supplementary to C Conclusion: ABCD is a parallelogram A B (180-x)º xº xº D C Theorems Proving Parallelograms Theorem 6.9 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. B Given: AC bisects BD and BD bisects AC Conclusion: ABCD is a parallelogram A D C Theorems Proving Parallelograms Theorem 6.10 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram. B Given: ADBC, AD ll BC C Conclusion: ABCD is a parallelogram A D Six Ways to Prove Quadrilaterals are Parallelograms 1. 2. 3. 4. 5. 6. Summary Show both pairs of opposite sides are parallel. Show both pairs of opposite sides are congruent. Show both pairs of opposite angles are congruent. Show both diagonals bisect each other. Show one angle is supplementary to both consecutive angles. Show one pair of opposite sides are both congruent and parallel. Example #1 Given: PQT RST Prove: PQRS is a parallelogram Q P T S R Example #2 Given: ABCD is a parallelogram FE ll DC Prove: ABEF is a parallelogram D C F A E B Example #3 Given: HJKM is a parallelogram IJKLMH Prove: HIKL is a parallelogram H M I 1 L J 2 K Example #4 Given:12, IJKLMH Prove: HIKL is a parallelogram H M I 1 L J 2 K Example #5 Show that A(-1,2), B(3,2), C(1,-2) and D(-3,-2) are the vertices of a parallelogram by a. Showing both pair of opposite sides are parallel. b. Showing both pair of opposite sides are congruent. Example #6 Identify any quadrilateral that is a parallelogram. a. G(-3,1), H(4,1), I(3,6), J(-1,6) b. P(-2,2), Q(1,1), R(4,4), S(1,4) c. W(3,-1), X(4,2), Y(1,5), Z(0,2) Homework Page 342 # 2-7, 9-19, 45-47