6-4: Squares and Rhombi

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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

• 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
6x12
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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?

• 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?

1. Diagonals are perpendicular
2. Diagonals are congruent
3. Consecutive sides are congruent
4. Consecutive angles are supplementary
5. 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)