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Methods for Proving Quadrilaterals are Parallelograms

In this chapter, we explore the ways to prove that a quadrilateral is a parallelogram, focusing on the definition and various theorems. A parallelogram, by definition, has both pairs of opposite sides parallel, but other criteria can confirm this property. Theorems include: 1) Opposite sides are congruent (Theorem 5-4), 2) One pair of sides is both congruent and parallel (Theorem 5-5), 3) Opposite angles are congruent (Theorem 5-6), and 4) Diagonals bisect each other (Theorem 5-7). Use these principles to deduce if a quadrilateral meets the criteria for being a parallelogram.

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Methods for Proving Quadrilaterals are Parallelograms

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  1. CHAPTER 5:QUADRILATERALS 5-2: WAYS TO PROVE QUADRILATERALS ARE PARALLELGORAMS

  2. PARALLELOGRAMS • Remember that a parallelogram, by definition, is a quadrilateral with both pairs of opposite sides parallel. • Conversely, if both pairs of opposite sides of a quadrilateral are parallel, then the quadrilateral is a parallelogram.

  3. THEOREMS Besides proving both pairs of opposite sides parallel, there are other ways to prove that quadrilaterals are parallelograms: THEOREMS 5-4 through 5-7 Pg. 172 of the textbook

  4. THEOREM 5-4 THEOREM 5-4: If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram. PG. 173, CE 1

  5. THEOREM 5-5 THEOREM 5-5: If one pair of opposite sides of a quadrilateral are both congruent and parallel, then the quadrilateral is a parallelogram. PG. 173, CE 8

  6. THEOREM 5-6 THEOREM 5-6: If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram. PG. 173, CE 6

  7. THEOREM 5-7 THEOREM 5-7: If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. PG. 173, CE 4

  8. If the measures of two angles of a quadrilateral are equal, then the quadrilateral is ________ a parallelogram. If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is ________ a parallelogram. To prove a quadrilateral is a parallelogram, it is ________ enough to show that one pair of opposite sides is parallel. Sometimes Always Never ALWAYS, SOMETIMES, NEVER

  9. State the definition or theorem that enables you to deduce, from the given information, that quadrilateral ABCD is a parallelogram: • BE = ED; CE = EA If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram. (Theorem 5-7) B C E A D

  10. CLASSWORK/HOMEWORK • CW: Pg. 173, Classroom Exercises 1-9, 12-13 • HW: Pg. 174, Written Exercises 1-10

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