Chapter 5 Quadrilaterals

1. Chapter 5Quadrilaterals • Apply the definition of a parallelogram • Prove that certain quadrilaterals are parallelograms • Apply the theorems and definitions about the special quadrilaterals

2. 5-1 Properties of Parallelograms Objectives • Apply the definition of a parallelogram • List the other properties of a parallelogram through new theorems

3. Quadrilaterals • Any 4 sided figure

4. If the opposite sides of a quadrilateral are parallel, then it is a parallelogram. A B D C Definition of a Parallelogram ( ) ABCD

5. Use the symbol for parallelogram and name using the 4 vertices in order either clockwise or counter clockwise. A B D C Naming a Parallelogram ABCD

6. Opposite sides of a parallelogram are congruent. Theorem A B D C

7. Opposite angles of a parallelogram are congruent. Theorem A B D C

8. The diagonals of a parallelogram bisect each other. Theorem A B D C

9. Remote Time • True or False

10. True or False • Every parallelogram is a quadrilateral

11. True or False • Every quadrilateral is a parallelogram

12. True or False • All angles of a parallelogram are congruent

13. True or False • All sides of a parallelogram are congruent

14. True or False • In RSTU, RS | |TU.  Hint draw a picture

15. True or False • In ABCD, if m  A = 50, then m  C = 130.  Hint draw a picture

16. True or False • In XWYZ, XY WZ  Hint draw a picture

17. True or False • In ABCD, AC and BD bisect each other  Hint draw a picture

18. White Board Practice Given ABCD Name all pairs of parallel sides

19. White Board Practice Given ABCD AB || DC BC || AD

20. White Board Practice Given ABCD Name all pairs of congruent angles

21. White Board Practice Given ABCD  BAD   DCB  CBD   ADB  ABC   CDA  ABD   CDB  BEA   DEC  BCA   DAC  BEC   DEA  BAC   DCA

22. White Board Practice Given ABCD Name all pairs of congruent segments

23. White Board Practice Given ABCD AB  CD BC  DA BE  ED AE  EC

24. White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. 6 R S yº xº 9 b 80º T U a

25. White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. x = 80 y = 45 a = 6 b = 9

26. White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. R S yº xº 9 12 a b 45º 35º U T

27. White Board Groups • Quadrilateral RSTU is a parallelogram. Find the values of x, y, a, and b. x = 100 y = 45 a = 12 b = 9

28. White Board Groups • Given this parallelogram with the diagonals drawn. 22 18 4y - 2 2x + 8

29. White Board Groups • Given this parallelogram with the diagonals drawn. x = 5 y = 6

30. 5-2:Ways to Prove that Quadrilaterals are Parallelograms Objectives • Learn about ways to prove a quadrilateral is a parallelogram

31. Show that both pairs of opposite sides of a quadrilateral are parallel Then the quadrilateral is a parallelogram A B D C Use the Definition of a Parallelogram

32. Show that both pairs of opposite sides are congruent. If both pairs of opposite sides of a quadrilateral are congruent, then it is a parallelogram. Theorem A B D C

33. Show that one pair of opposite sides are both congruent and parallel. If one pair of opposite sides of a quadrilateral are both congruent and parallel, then it is a parallelogram. Theorem A B D C

34. Show that both pairs of opposite angles are congruent. If both pairs of opposite angles of a quadrilateral are congruent, then it is a parallelogram. Theorem A B D C

35. Show that the diagonals bisect each other. If the diagonals of a quadrilateral bisect each other, then it is a parallelogram. Theorem A B X D C

36. Five ways to prove a Quadrilateral is a Parallelogram • Show that both pairs of opposite sides parallel • Show that both pairs of opposite sides congruent • Show that one pair of opposite sides are both congruent and parallel • Show that both pairs of opposite angles congruent • Show that diagonals that bisect each other

37. The diagonals of a quadrilateral _____________ bisect each other A. Sometimes • Always • Never • I don’t know

38. If the measure of two angles of a quadrilateral are equal, then the quadrilateral is ____________ a parallelogram • Sometimes • Always • Never • I don’t know

39. If one pair of opposite sides of a quadrilateral is congruent and parallel, then the quadrilateral is ___________ a parallelogram A. Sometimes B. Always C. Never D. I don’t know

40. If both pairs of opposite sides of a quadrilateral are congruent, then the quadrilateral is __________ a parallelogram A.) Sometimes B.) Always C.) Never D.) I don’t know

41. To prove a quadrilateral is a parallelogram, it is ________ enough to show that one pair of opposite sides is parallel. A.) Sometimes B.) Always C.) Never D.) I don’t know

42. 5-3 Theorems Involving Parallel Lines Objectives • Apply the theorems about parallel lines and triangles

43. If two lines are parallel, then all points on one line are equidistant from the other. Theorem m n

44. If three parallel lines cut off congruent segments on one transversal, then they do so on any transversal. Theorem A D B E C F

45. A line that contains the midpoint of one side of a triangle and is parallel to a another sidepasses through the midpoint of the third side. Theorem A X Y B C

46. A segment that joins the midpoints of two sides of a triangleis parallel to the third side and its length is half the length of the third side. Theorem A X Y B C

47. White Board Practice • Given: R, S, and T are midpoint of the sides of  ABC B R S C A T

48. White Board Practice • Given: R, S, and T are midpoint of the sides of  ABC B R S C A T

49. White Board Practice • Given that AR | | BS | | CT; RS  ST R S A T B C

50. White Board Practice • Given that AR | | BS | | CT; RS  ST If RS = 12, then ST = ____ R S A T B C