1. 8.2/8.3 Parallelograms

2. You will learn to identify and use the properties of parallelograms. 1) Parallelogram

3. A parallelogram is a quadrilateral with two pairs of ____________. parallel sides A B D C In parallelogram ABCD below, and Also, the parallel sides are _________. congruent Knowledge gained about “parallels” (chapter 4) will now be used in the following theorems.

4. Theorem 8-2 Theorem 8-3 Theorem 8-4 Opposite angles of a parallelogram are ________. Opposite sides of a parallelogram are ________. The consecutive angles of a parallelogram are ____________. A B D C A B D C A B D C  A   C and  B   D m  A + m  B = 180 m  D + m  C = 180 congruent supplementary

5. In RSTU, RS = 45, ST = 70, and  U = 68. R U S T 45 70 68° Find: RU = ____ UT = _____ m  S = _____ m  T = _____ 70 Theorem 8-3 45 Theorem 8-3 68° Theorem 8-2 112° Theorem 8-4

6. Theorem 8-5 The diagonals of a parallelogram ______ each other. A D B C E bisect In RSTU, if RT = 56, find RE. R U S T E RE = 28

7. A D B C In the figure below, ABCD is a parallelogram. DB  BD Since AD || BC and diagonal DB is a transversal, then  ADB   CBD. (Alternate Interior angles) Since AB || DC and diagonal DB is a transversal, then  BDC   DBA. (Alternate Interior angles) ASA Theorem

8. Theorem 8-6 A diagonal of a parallelogram separates it into two _________________. A D B C congruent triangles

9. The Escher design below is based on a _____________. You can use a parallelogram to make a simple Escher-like drawing. Change one side of the parallelogram and then translate (slide) the change to the opposite side. The resulting figure is used to make a design with different colors and textures. parallelogram

11. You will learn to identify and use tests to show that a quadrilateral is a parallelogram. Nothing New!

12. Theorem 8-7 If both pairs of opposite sides of a quadrilateral are _________, then the quadrilateral is a parallelogram. A D C B congruent

13. You can use the properties of congruent triangles and Theorem 8-7 to find other ways to show that a quadrilateral is a parallelogram. In quadrilateral PQRS, PR and QS bisect each other at T. Show that PQRS is a parallelogram by providing a reason for each step. Definition of segment bisector Vertical angles are congruent SAS Corresp. parts of Congruent Triangles are Congruent Theorem 8-7 T P S R Q

14. Theorem 8-8 If one pair of opposite sides of a quadrilateral is _______ and _________, then the quadrilateral is a parallelogram. A D C B congruent parallel

15. Theorem 8-9 If the diagonals of a quadrilateral ________________, then the quadrilateral is a parallelogram. bisect each other A DC B E

16. Determine whether each quadrilateral is a parallelogram. If the figure is a parallelogram, give a reason for your answer. A D C B Given Alt. Int. Angles Therefore, quadrilateral ABCD is a parallelogram. Theorem 8-8