2. Aim: Properties of Parallelogram Course: Applied Geo. Do Now: Aim: What are the Properties of a Parallelogram? Describe the properties of an isosceles right triangle.

3. Aim: Properties of Parallelogram Course: Applied Geo. Both pairs of opposite sides are parallel AB || CD & AC || BD A parallelogram is a quadrilateral with special features Opposite angles are congruent C A B D  A   D &  B   C Consecutive angles are supplementary  A +  B = 180 0,  B +  D = 180 0 etc. The sum of the four angles is 360 0  A +  B +  C +  D = 360 0 Opposite sides are congruent AB  CD & AC  BD What is a parallelogram?

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

5. Aim: Properties of Parallelogram Course: Applied Geo. In the diagram below, ABCD is a parallelogram, If m  C = 106 0, find the m  A, m  B and m  D AD BC 106 0  A and  C are opposite angles in a parallelogram and must be equal measures.  A = 106 0  B is supplementary to  C. Therefore  B = 74 0  B and  D are opposite angles and must be equal measures.  D = 74 0 106 0 74 0 74 0 Model Problem

6. Aim: Properties of Parallelogram Course: Applied Geo. A D BC 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. 3x – 33 2x + 28  A and  B are consecutive angles of a parallelogram and are therefore supplementary. (2x + 28) + (3x – 33) = 180 5x – 5 = 180 5x = 185 x = 37  B &  D = 2(37) + 28 = 102 0  C &  A = 3(37) – 33 = 78 0 78 0 102 0 Model Problem

7. Aim: Properties of Parallelogram Course: Applied Geo. In parallelogram PQRS, PQ = 12y – 14 and SR = 3y + 49. Find the value of y, PQ and SR. QP RS 12y – 14 3y + 49 Opposite sides of a parallelogram are congruent. Therefore: 3y + 49 = 12y – 14 -3y 49 = 9y – 14 63 = 9y 7 = y 3(7) + 49 = 70 12(7) – 14 = 70 Model Problem

8. Aim: Properties of Parallelogram Course: Applied Geo. Find the value of the variables. Model Problem Opposite angles are congruent. Consecutive angles are supplementary. In a parallelogram: 3y + y = 180 4y = 180 y = 45 3y = 135 3x = y 3x = 45 x = 15 45 135 45 y 3x3x 3y3y

9. Aim: Properties of Parallelogram Course: Applied Geo. DB & AC are diagonals of ABCD. AD B C Explain how  DEA   BCE? E Informal Proofs with Parallelograms  CEB   AED – Vertical angles  BCE   EAD – CB | | DA forming alternate interior angles that are congruent  DEA   BCE because of AAS  AAS (A  A) CB  AD – Opposite sides of parallelogram are congruent (S  S) SAS  SAS SSS  SSS ASA  ASA AAS  AAS HL  HL

10. Aim: Properties of Parallelogram Course: Applied Geo. AD B C Name the corresponding parts of  DEA &  BCE E 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 More about Diagonals of Parallelograms

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

12. Aim: Properties of Parallelogram Course: Applied Geo. 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 = Model Problem A B D C 30 o 115 o 65 o 35 o 30 o 115 o 65 o 30 o 35 o

13. Aim: Properties of Parallelogram Course: Applied Geo. Find the values for x and y in parallelogram QRST Since the diagonals of a parallelogram bisect each other the following is true: y + 6 = 3y - 44x + 1 = 3x + 4 -y 6 = 2y - 4 +4 + 4 10 = 2y 5 = y -3x x + 1 = + 4 - 1 = - 1 x = 3 Model Problem QT RS y + 6 3y - 4 4x + 1 3x + 4