Proving Quadrilaterals are Parallelograms

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Proving Quadrilaterals are Parallelograms

• Lesson 5.2
• Pre-AP Geometry

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Objective

• Prove that certain quadrilaterals are
  parallelograms.

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Quadrilaterals and Parallelograms

• A quadrilateral is a polygon with 4 sides.
• A parallelogram is a quadrilateral whose opposite
  sides are parallel (the top and bottom are
  parallel and the left and right are parallel).

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Theorem 5-4

• If both pairs of opposite sides of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

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Theorem 5-5

• If one pair of opposite sides of a quadrilateral
  are both congruent and parallel, then the
  quadrilateral is a parallelogram.

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Theorem 5-6

• If both pairs of opposite angles of a
  quadrilateral are congruent, then the
  quadrilateral is a parallelogram.

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Theorem 5-7

• If the diagonals of a quadrilateral bisect each
  other, then the quadrilateral is a parallelogram.

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Five ways to Prove that a Quadrilateral is a
Parallelogram

• Show that both pairs of opposite sides are
  parallel.
• Show that both pairs of opposite sides are
  congruent.
• Show that one pair of opposite sides are both
  congruent and parallel.
• Show that both pairs of opposite angles are
  congruent.
• Show that the diagonals bisect each other.

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Review

• Answer with always, sometimes, or never.
• The diagonals of a quadrilateral bisect each
  other.
• 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
  is congruent and parallel, then the quadrilateral
  is a parallelogram.
• If both pairs of opposite sides of a
  quadrilateral are congruent, 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.

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Practice

• State the definition or theorem that enables you
  to deduce, from the information provided, that
  quadrilateral ABCD is a parallelogram.
• BE ED CE EA
• ?BAD ? ?DCB ?ADC ? ?CBA
• BC AD AB DC
• BC ? AD AB ? DC

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Written Exercises

• Problem Set 5.2, p. 174 1 10, 12, 19 - 22