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Aim: What are the Properties of a Parallelogram?

Aim: What are the Properties of a Parallelogram?. Do Now:. Describe the properties of an isosceles right triangle. What is a parallelogram?. B. A. A parallelogram is a quadrilateral with special features. C. D. Both pairs of opposite sides are parallel. AB || CD & AC || BD.

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Aim: What are the Properties of a Parallelogram?

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  1. Aim: What are the Properties of a Parallelogram? Do Now: Describe the properties of an isosceles right triangle.

  2. What is a parallelogram? B A A parallelogram is a quadrilateral with special features C D Both pairs of opposite sides are parallel AB || CD & AC || BD Opposite angles are congruent A  D & B  C Consecutive angles are supplementary A + B = 1800, B + D = 1800 etc. The sum of the four angles is 3600 A + B + C + D = 3600 Opposite sides are congruent AB  CD & AC  BD

  3. Definition of a Parallelogram B A A parallelogram is a quadrilateral in which two pairs of opposite sides are parallel. C D Opposite sides are parallel. Opposite sides are congruent. Opposite angles are congruent. Consecutive angles are supplementary. Diagonals bisect each other. A diagonal divides a parallelogram into two congruent triangles.

  4. Model Problem In the diagram below, ABCD is a parallelogram, If mC = 1060, find the mA, mB and mD C B 740 1060 740 1060 D A A and C are opposite angles in a parallelogram and must be equal measures. A = 1060 B is supplementary to C. Therefore B = 740 B and D are opposite angles and must be equal measures. D = 740

  5. Model Problem In parallelogram ABCD, mA = 3x – 33, and B = 2x + 28. Find the value of x and the measure of all four angles of the parallelogram. C B 2x + 28 780 3x – 33 1020 A D A and B are consecutive angles of a parallelogram and are therefore supplementary. (2x + 28) + (3x – 33) = 180 5x – 5 = 180 B & D = 2(37) + 28 = 1020 5x = 185 C & A = 3(37) – 33 = 780 x = 37

  6. Model Problem In parallelogram PQRS, PQ = 12y– 14 and SR = 3y + 49. Find the value of y, PQ and SR. S R 3y + 49 3(7) + 49 = 70 12(7) – 14 = 70 12y– 14 P Q Opposite sides of a parallelogram are congruent. Therefore: 3y + 49 = 12y– 14 -3y-3y 49 = 9y– 14 63 = 9y 7 = y

  7. Model Problem Find the value of the variables. y 45 45 3x 3y 135 In a parallelogram: Consecutive angles are supplementary. Opposite angles are congruent. 3y + y = 180 3x = y 3x = 45 4y = 180 y = 45 x = 15 3y = 135

  8. SAS  SAS SSS  SSS ASA  ASA AAS  AAS HL  HL Informal Proofs with Parallelograms DB & AC are diagonals of ABCD. B C E D A Explain how DDEA DBCE? BCE EAD –CB | | DA forming alternate interior angles that are congruent (A  A) CEB AED – Vertical angles (A  A) CB AD – Opposite sides of parallelogram are congruent (S  S) DEA BCEbecause ofAAS  AAS

  9. More about Diagonals of Parallelograms B C E D A Name the corresponding parts of DDEA &DBCE CE  AE DE  BE DA  BC CEBAED BCEDAE CBEEDA What is true about the diagonals of a parallelogram based on the congruence of the two triangles? The diagonals of a parallelogram bisect each other. DE  EB and CE  AE

  10. B C E A D Diagonals of a Parallelogram The diagonals of a parallelogram bisect each other. In parallelogram ABCD with diagonals AC and BD, AC = 2x – 14 and EC = 20. Find the value of x. AC = 2(EC) check: AC = 2x – 14 2x – 14 = 2(20) = 40 AC = 2(27) – 14 2x = 54 x = 27 AC = 40

  11. Model Problem Parallelogram ABCD is cut by a diagonal AC. The mCAD = 30 and the mB = 115. Find the measure of the following angles. mD = mBAD = mBCD = mBCA = mBAC = mACD = 65o 115o B C 30o 115o 65o 35o 35o 65o 115o 30o A D 65o 30o 35o 35o

  12. Model Problem Find the values for x and y in parallelogram QRST S R y + 6 3y - 4 4x + 1 3x + 4 T Q Since the diagonals of a parallelogram bisect each other the following is true: y + 6 = 3y - 4 4x + 1 = 3x + 4 -y-y -3x-3x 6 = 2y - 4 x + 1 = + 4 +4+ 4 - 1 = - 1 10 = 2y x = 3 5 = y

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