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Geometry Lesson 64 Special Parallelograms.notebook November 05, 2012 Section 64 : Special Parallelograms WarmUp PACE is a parallelogram and m<PAC = 109. Complete each of the following. 7 A C 3. 5 1. EC = 5 4.5 4.7 2. EP = R 3. m<CEP = P E 4. PR = 5. RE = 6. CP = 7. m<EPA = 8. m<ECA = If you draw two congruent isosceles triangles with base PQ, you have drawn a rhombus. R Note that <RPQ, <SPQ, <RQP, and <SQP are all congruent This suggests Theorem 69. Q P S Theorem 69: Each diagonal of a rhombus bisects two angles of the rhombus. Proof: 1 Geometry Lesson 64 Special Parallelograms.notebook November 05, 2012 Theorem 610: The diagonals of a rhombus are perpendicular. B C AC BD A D Theorem 611: The diagonals of a rectangle are congruent. B C A D AC ≅ BD Example 1) MNPQ is a rhombus and m<N = 120. Find the measures of the numbered angles. N P 3 120 o 4 1 M 2 Q 2 Geometry Lesson 64 Special Parallelograms.notebook November 05, 2012 You try) Find the measures of the numbered angles in the rhombus. 50o 1 4 2 3 Example 2) Find the length of the diagonals of rectangle GFED if FD = 2y + 4 and GE = 6y 5. F E G D 3 Geometry Lesson 64 Special Parallelograms.notebook You try) Find the lengths of the diagonals of GFED if FD = 5y 9 and GE = y + 5 F G November 05, 2012 E D Theorem 612: If one diagonal of a parallelogram bisects two angles of the parallelogram, then the parallelogram is a rhombus. Theorem 613: If the diagonals of a parallelogram are perpendicular, then the parallelogram is a rhombus. Theorem 614: If the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle. 4 Geometry Lesson 64 Special Parallelograms.notebook November 05, 2012 Example 3) Determine whether the quadrilateral can be a parallelogram. If not, write impossible. a. The quadrilateral has congruent diagonals and one angle of 60 o. b. The quadrilateral has perpendicular diagonals and four right angles. You try) A diagonal of a parallelogram bisects two angles of the parallelogram. It is possible for the parallelogram to have sides of lengths 5, 6, 5, and 6? Explain. 5 Geometry Lesson 64 Special Parallelograms.notebook November 05, 2012 Example 4) Builders use properties of diagonals to "square up" rectangular shapes like building frames and playingfield boundaries. Suppose you are on the volunteer building team. You are helping to lay out a rectangular patio. Explain how to use 2 ropes and the properties of diagonals to locate the four corners. 6