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6.4 Special Parallelograms

6.4 Special Parallelograms. Definitions. Rhombus – a parallelogram with four congruent sides. Rectangle – a parallelogram with four right angles. Rhombuses Results. Theorem 6-9: Each diagonal of a rhombus bisects two angles of the rhombus.

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6.4 Special Parallelograms

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  1. 6.4 Special Parallelograms

  2. Definitions • Rhombus – a parallelogram with four congruent sides. • Rectangle – a parallelogram with four right angles.

  3. Rhombuses Results • Theorem 6-9: Each diagonal of a rhombus bisects two angles of the rhombus. • Theorem 6-10: The diagonals of a rhombus are perpendicular.

  4. Rectangles Results • Theorem 6.11: The diagonals of a rectangle are congruent.

  5. Example 1 • Find the measures of the numbered angles in the rhombus. 1 2 78° 2 3 4

  6. Example 2 • One diagonal of a rectangle has length 8x + 2. The other diagonal has length 5x + 11. Find the length of each diagonal.

  7. Homework • Page 315 # 1-15

  8. Rhombus or Rectangle? • Theorem 6.12: If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. • Theorem 6.13: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. • Theorem 6.14: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

  9. Example 1 • A diagonal of a parallelogram bisects two angles of the parallelogram. Is it possible for the parallelogram to have sides of lengths 5, 6, 5, and 6? Explain.

  10. Example 2 • The diagonals of ABCD are perpendicular. AB = 16 cm and BC = 8 cm. Can ABCD be a parallelogram? Explain.

  11. Example 3 • Builders use properties of diagonals to “square up” rectangular shapes like building frames and playing field boundaries. • How could you use diagonals to locate the four corners of a rectangular patio? • How could you use diagonals to locate the four corners of a square patio? • How could you use diagonals to locate the for corners of a patio shaped like a rhombus?

  12. Homework • Page 315 # 16-21, 45-53

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