6-4: Squares and Rhombi

Expectations:G1.4.1: Solve multistep problems and construct

proofs involving angle measure, side length, diagonal length, perimeter, and area of squares, rectangles, parallelograms, kites, and trapezoids.

G1.4.2: Solve multistep problems and construct proofs involving quadrilaterals (e.g., prove that the diagonals of a rhombus are perpendicular) using Euclidean methods or coordinate geometry.

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Rhombus

Defn: Rhombus: A quadrilateral is a rhombus iff all 4 sides are congruent. The plural or rhombus is rhombi.

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Properties of a Rhombus Theorem

If a quadrilateral is a rhombus, then:

a.it is a parallelogram.

b. the diagonals are perpendicular to each other.

c. each diagonal bisects a pair of opposite angles.

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Prove a rhombus is a parallelogram.

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The figure below is a rhombus. Solve for x.

10x - 24

6x+12

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Sufficient Condition for a Rhombus Theorem

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

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Determine the value of x so that the parallelogram is a rhombus.

(15x – 30)

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Square

Defn: Square: A parallelogram is a square iff it is a rectangle and a rhombus.

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What is true about the diagonals of a square?

a.congruent (rectangle),

b. perpendicular (rhombus),

c. bisect a pair of opposite angles (rhombus),

d. bisect each other (parallelogram)

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WXYZ is a quadrilateral. Of the terms parallelogram, rectangle, rhombus, square which apply to WXYZ?

W(5,5), X(10,5), Y(10,10), Z(5,10)

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Which of the following is a property of squares, but not rhombi?

A) Diagonals are perpendicularB) Diagonals are congruentC) Consecutive sides are congruentD) Consecutive angles are supplementaryE) Opposite angles are congruent

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Prove the diagonals of a square are congruent.

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Assignment

Pages 317 – 318,# 21 – 35, 39 – 47 (odds)

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