6'4 Rhombuses, Rectangles, and Squares 13108

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6.4 Rhombuses, Rectangles, and Squares1/31/08

• Three special types of parallelograms
• Rhombus
• Rectangle
• Square

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Rhombus

• Rhombus a parallelogram with four congruent
  sides.

Rhombus
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Rectangle

• Rectangle a parallelogram with four right
  angles.

Rectangle
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Square

• Square a parallelogram with four congruent
  sides and four right angles.

Square
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Use Properties of Special Parallelograms

• In the diagram, ABCD is a rectangle.
• a. Find AD and AB.
• b. Find m A, m B, m C, and m
  D.
• AD BC 5, and AB DC 8.
• By definition, a rectangle has 4 right angles, so
  
• m A m B m C m D 90.

A
B
5
D
C
8
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Rhombus Corollary

• If a quadrilateral has four congruent sides, then
  it is a rhombus.
• If AB BC CD AD, then ABCD is a rhombus.

A
B
C
D
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Rectangle Corollary

• If a quadrilateral has four right angles, then it
  is a rectangle.
• If m A m B m C m D 90,
  then ABCD is a rectangle.

A
B
D
C
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Square Corollary

• If a quadrilateral has four congruent sides and
  four right angles, then it is a square.
• If AB BC CD AD and m A m B m
  C m D 90, then ABCD is a square.

A
B
D
C
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Identify Special Quadrilaterals

• Use the information in the diagram to name the
  quadrilateral.
• The quadrilateral has four right angles. So, by
  the Rectangle Corollary, the quadrilateral is a
  rectangle.
• Because all of the sides are not the same length,
  you know that the quadrilateral is not a square.

3
5
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Theorem 6.10

• The diagonals of a rhombus are perpendicular.
• In rhombus ABCD, AC BD.

B
C
D
A
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Use Diagonals of a Rhombus

• ABCD is a rhombus. Find the value of x.
• By Theorem 6.10, the diagonals of a rhombus are
  perpendicular. Therefore, BEC is a right
  angle, so BEC is a right triangle.
• So, x 90 60 30.

B
C
x
60
E
D
A
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Theorem 6.11

• The diagonals of a rectangle are congruent.
• In rectangle ABCD, AC BD.

A
B
D
C
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More Practice!!!

• Textbook p. 328 2 9, 14 20