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Honors Geometry Section 4.6 (1) Conditions for Special Quadrilaterals

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In section 4.5, we answered questions such as “If a quadrilateral is a parallelogram, what are its properties?” or “If a quadrilateral is a rhombus, what are its properties?” In this section we look to reverse the process, and answer the question “What must we know about a quadrilateral in order to say it is a parallelogram or a rectangle or a whatever?”

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What does it take to make a parallelogram? State whether the following conjectures are true or false. If it is false, draw a counterexample.

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

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

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

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

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

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

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The last 4 statements will be our tests for determining if a quadrilateral is a parallelogram. If a quadrilateral does not satisfy one of these 4 tests, then we cannot say that it is a parallelogram!

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What does it take to make a rectangle? State whether the following conjectures are true or false. If it is false, draw a counterexample.

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If one angle of a quadrilateral is a right angle, then the quadrilateral is a rectangle.

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If one angle of a parallelogram is a right angle, then the parallelogram is a rectangle.

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If the diagonals of a quadrilateral are congruent, then the quadrilateral is a rectangle.

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If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

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If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rectangle.

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Statements 2 and 4 will be our tests for determining if a quadrilateral is a rectangle. Notice that in both of those statements you must know that the quadrilateral is a parallelogram before you can say that it is a rectangle.

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What does it take to make a rhombus? State whether the following conjectures are true or false. If it is false, draw a counterexample.

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If one pair of adjacent sides of a quadrilateral are congruent, then the quadrilateral is a rhombus.

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If one pair of adjacent sides of a parallelogram are congruent, then the parallelogram is a rhombus.

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If the diagonals of a parallelogram are congruent, then the parallelogram is a rhombus.

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If the diagonals of a parallelogram are perpendicular then the parallelogram is a rhombus.

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If the diagonals of a parallelogram bisect the angles of the parallelogram, then the parallelogram is a rhombus.

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Statements 2, 4 and 5will be our tests for determining if a quadrilateral is a rhombus. Notice that in each of these statements you must know that the quadrilateral is a parallelogram before you can say that it is a rhombus.

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What does it take to make a square? It must be a parallelogram, rectangle and rhombus.

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Examples: Consider quad. OHMY with diagonals that intersect at point S. Determine if the given information allows you to conclude that quad. OHMY is a parallelogram, rectangle, rhombus or square. List all that apply.

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