64 Squares and Rhombi

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6-4Squares and Rhombi

• By Shannon Adison
• Mr. Hales Geometry
• 2nd period

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Rhombi

• The optical illusion is created using special
  quadrilaterals called Rhombi.
• A Rhombus is a quadrilateral with four congruent
  sides. Since opposite sides of a rhombus are
  congruent, a rhombus is a parallelogram.

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Theorems

• Theorem 6-11 The diagonals of a rhombus are
  perpendicular.
• Theorem 6-12 If the diagonals of a parallelogram
  are perpendicular, then the parallelogram is a
  rhombus.
• Theorem 6-13 Each diagonal of a rhombus bisects
  a pair of opposite angles.

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Proof of Theorem 6-13Prove Each diagonal
bisects a pair of opposite angles

• 1. ABCD is a rhombus
• 2.ABCD is a parallelogram
• 3. ABCD ADC, BAD BCD,
• 4.ABBCCDDA
• 5. ABC ADC
• 6. 5 6, 7 8
• 7. BAD BCD
• 8. 1 2, 3 4
• 9. Each diagonal bisects a pair of opposite
  angles.

• 1. Given
• 2. Def.rhombus
• 3. Opp. Angles of a are
• 4. Def. rhombus
• 5.SAS
• 6. CPCTC
• 7. SAS
• 8. CPCTC
• 9. Def. of Angle bisector

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Squares

• If a quadrilateral is both a rhombus and a
  rectangle, it is a square.
• A square is a quadrilateral with four right
  angles and four congruent sides.
• You can construct a square in the same way that
  you constructed rhombus.

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Examples

• You can use the properties of a rhombus to solve
  problems.
• A if m 12x20 and m 2 5x-4, find the value
  of x.
• The diagonals of a rhombus bisect a pair of
  opposite angles. So, CE bisects BCD and m 1m
  2.
• m 1m 2 B 1 C
• 2x205x-4 3 2
• 243x f
• 8x E D

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• B If BD15, find BF.
• Since a rhombus is a parallelogram, its diagonals
  bisect each other.
• Thus, if BD15, BF1/2 (15) or 7.5.
• C If m 3y226, find y.
• The diagonals of a rhombus are perpendicular, so
  3 is a right angle .
• m 390 B 1 C
• y22690 3 2
• y264 E
• y8 or -8 E
  D