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6 – 4 Special Parallelograms

6 – 4 Special Parallelograms. L.E.Q. How do you use properties of diagonals of rhombuses and rectangles and determine whether a parallelogram is a rhombus or a rectangle?. Theorem 6 – 9: Rhombus Diagonals Bisect Angles. 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 L.E.Q. How do you use properties of diagonals of rhombuses and rectangles and determine whether a parallelogram is a rhombus or a rectangle?

  2. Theorem 6 – 9: Rhombus Diagonals Bisect Angles. • Each diagonal of a rhombus bisects two angles of the rhombus.

  3. Theorem 6 – 10: Rhombus Diagonals are Perpendicular. • The diagonals of a rhombus are perpendicular.

  4. Example 1: Finding Angle Measures. • MNPQ is a rhombus and • Find the measure of the numbered angles.

  5. Find the measures of the numbered angles in the rhombus below.

  6. Theorem 6 – 11: Diagonals of a Rectangle. • The diagonals of a rectangle are congruent.

  7. Example 2: Finding Diagonal Length. • Find the length of the diagonals of rectangle GFED if FD = 2y + 4 and GE = 6y – 5.

  8. Find the length of the diagonals of GFED if FD = 5y - 9 and GE = y + 5.

  9. Theorems Used to Prove Whether a Parallelogram is a Rhombus or a Rectangle.

  10. Theorem 6 – 12: • If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus.

  11. Theorem 6 – 13: • If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus.

  12. Theorem 6 – 14: • If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.

  13. Example 3: Recognizing Special Parallelograms. • Determine whether the quadrilateral can be a parallelogram. If not, write impossible. Explain. • The quadrilateral has congruent diagonals and one angle of 60 degrees. • The quadrilateral has perpendicular diagonals and four right angles.

  14. 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.

  15. Example 4: Real-World Connection. • Community Service Builders use properties of diagonals to “square up” rectangular shapes like building frames and playing-field boundaries. Suppose you are on the volunteer building team at the right. You are helping to lay out a rectangular patio. Explain how to use properties of diagonals to locate the four corners.

  16. Homework: • Pgs. 315-317 #s 2 – 20 even, 48 – 50 all.

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