1. Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems

2. Proving Lines Parallel

3. Proving Triangles Congruent

5. Four Window Foldable Start by folding a blank piece of paper in half lengthwise, and then folding it in half in the opposite direction. Now fold it in half one more time in the same direction.

6. Four Window Foldable Now unfold the paper, and then while holding the paper vertically, fold down the top one-fourth to meet the middle. Do the same with the bottom one- fourth.

7. Four Window Foldable To finish your foldable, cut the two vertical fold lines to create four windows. Outside: Outside: Property 1-4 Inside Flap: Inside Flap: Illustration Inside: Inside: Theorem

8. Investigation 1 In this lesson, we will find ways to show that a quadrilateral is a parallelogram. Obviously, if the opposite sides are parallel, then the quadrilateral is a parallelogram. But could we use other properties besides the definition to see if a shape is a parallelogram?

9. 8.3 Show a Quadrilateral is a Parallelogram Objectives: 1.To use properties to identify parallelograms

10. Property 1 We know that the opposite sides of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite sides are congruent, then is it also a parallelogram? Step 1: Draw a quadrilateral with congruent opposite sides.

11. Property 1 Step 2: Draw diagonal AD. Notice this creates two triangles. What kind of triangles are they? by SSS 

12. Property 1 Step 3: Since the two triangles are congruent, what must be true about  BDA and  CAD? by CPCTC

13. Property 1 Step 4: Now consider AD to be a transversal. What must be true about BD and AC? by Converse of Alternate Interior Angles Theorem

14. Property 1 Step 5: By a similar argument, what must be true about AB and CD? by Converse of Alternate Interior Angles Theorem

15. Property 1 If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

16. Property 2 We know that the opposite angles of a parallelogram are congruent. What about the converse? If we had a quadrilateral whose opposite angles are congruent, then is it also a parallelogram? Step 1: Draw a quadrilateral with congruent opposite angles.

17. Property 2 Step 2: Now assign the congruent angles variables x and y. What is the sum of all the angles? What is the sum of x and y ?

18. Property 2 Step 3: Consider AB to be a transversal. Since x and y are supplementary, what must be true about BD and AC? by Converse of Consecutive Interior Angles Theorem

19. Property 2 Step 4: By a similar argument, what must be true about AB and CD? by Converse of Consecutive Interior Angles Theorem

20. Property 2 If both pairs of opposite angles of a quadrilateral are congruent, then the quadrilateral is a parallelogram.

21. Property 3 We know that the diagonals of a parallelogram bisect each other. What about the converse? If we had a quadrilateral whose diagonals bisect each other, then is it also a parallelogram? Step 1: Draw a quadrilateral with diagonals that bisect each other.

22. Property 3 Step 2: What kind of angles are  BEA and  CED? So what must be true about them? by Vertical Angles Congruence Theorem

23. Property 3 Step 3: Now what must be true about AB and CD? by SAS  and CPCTC

24. Property 3 Step 4: By a similar argument, what must be true about BD and AC? by SAS  and CPCTC

25. Property 3 Step 5: Finally, if the opposite sides of our quadrilateral are congruent, what must be true about our quadrilateral? ABDC is a parallelogram by Property 1

26. Property 3 If the diagonals of a quadrilateral bisect each other, then the quadrilateral is a parallelogram.

27. Property 4 The last property is not a converse, and it is not obvious. The question is, if we had a quadrilateral with one pair of sides that are congruent and parallel, then is it also a parallelogram? Step 1: Draw a quadrilateral with one pair of parallel and congruent sides.

28. Property 4 Step 2: Now draw in diagonal AD. Consider AD to be a transversal. What must be true about  BDA and  CAD? by Alternate Interior Angles Theorem

29. Property 4 Step 3: What must be true about  ABD and  DCA? What must be true about AB and CD? by SAS and CPCTC

30. Property 4 Step 4: Finally, since the opposite sides of our quadrilateral are congruent, what must be true about our quadrilateral? ABDC is a parallelogram by Property 1

31. Property 4 If one pair of opposite sides of a quadrilateral are congruent and parallel, then the quadrilateral is a parallelogram.

32. Example 1 In quadrilateral WXYZ, m  W = 42°, m  X = 138°, and m  Y = 42°. Find m  Z. Is WXYZ a parallelogram? Explain your reasoning.

33. Example 2 For what value of x is the quadrilateral below a parallelogram?

34. Example 3 Determine whether the following quadrilaterals are parallelograms.

35. Example 4 Construct a flowchart to prove that if a quadrilateral has congruent opposite sides, then it is a parallelogram. Given: AB  CD BC  AD Prove: ABCD is a parallelogram

36. Summary

37. Assignment P. 526-529: 1-11, 15-21, 33-36, 38, 41, 43 Challenge Problems