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1. Quadrilaterals Eleanor Roosevelt High School Chin-Sung Lin

2. ERHS Math Geometry Definitions of the Quadrilaterals Mr. Chin-Sung Lin

3. ERHS Math Geometry Quadrilaterals A quadrilateral is a polygon with four sides Mr. Chin-Sung Lin

4. ERHS Math Geometry Parts & Properties of the Quadrilaterals Mr. Chin-Sung Lin

5. ERHS Math Geometry Consecutive (Adjacent) Vertices Consecutive vertices or adjacent vertices are vertices that are endpoints of the same side P and Q, Q and R, R and S, S and P Q P R S Mr. Chin-Sung Lin

6. ERHS Math Geometry Consecutive (Adjacent) Sides Consecutive sides or adjacent sides are sides that have a common endpoint PQ and QR, QR and RS, RS and SP, SP and PQ Q P R S Mr. Chin-Sung Lin

7. ERHS Math Geometry Opposite Sides Opposite sides of a quadrilateral are sides that do not have a common endpoint PQ and RS, SP and QR Q P R S Mr. Chin-Sung Lin

8. ERHS Math Geometry Consecutive angles Consecutive angles of a quadrilateral are angles whose vertices are consecutive P and Q, Q and R, R and S, S and P Q P R S Mr. Chin-Sung Lin

9. ERHS Math Geometry Opposite Angles Opposite angles of a quadrilateral are angles whose vertices are not consecutive P and R, Q and S Q P R S Mr. Chin-Sung Lin

10. ERHS Math Geometry Diagonals A diagonal of a quadrilateral is a line segment whose endpoints are two nonadjacent vertices of the quadrilateral PR and QS Q P R S Mr. Chin-Sung Lin

11. ERHS Math Geometry Sum of the Measures of Angles The sum of the measures of the angles of a quadrilateral is 360 degrees mP + mQ + mR + mS = 360 Q P R S Mr. Chin-Sung Lin

12. ERHS Math Geometry Parallelograms Mr. Chin-Sung Lin

13. A B D C ERHS Math Geometry A parallelogram is a quadrilateral in which two pairs of opposite sides are parallel AB || CD, AD || BC A parallelogram can be denoted by the symbol ABCD The use of arrowheads, pointing in the same direction, to show sides that are parallel in the figure Parallelogram Mr. Chin-Sung Lin

14. ERHS Math Geometry Theorems of Parallelogram Mr. Chin-Sung Lin

15. ERHS Math Geometry Theorems of Parallelogram Theorem of Dividing Diagonals Theorem of Opposite Sides Theorem of Opposite Angles Theorem of Bisecting Diagonals Theorem of Consecutive Angles Mr. Chin-Sung Lin

16. A B D C ERHS Math Geometry • A diagonal divides a parallelogram into two congruent triangles • If ABCD is a parallelogram, then • ∆ ABD∆ CDB Theorem of Dividing Diagonals Mr. Chin-Sung Lin

17. 1 3 4 2 A B D C ERHS Math Geometry Theorem of Dividing Diagonals Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. 12 and 34 3. Alternate interior angles 4. BD BD 4. Reflexive property 5. ∆ ABD  ∆ CDB 5. ASA postulate Mr. Chin-Sung Lin

18. A B D C ERHS Math Geometry • Opposite sides of a parallelogram are congruent • If ABCD is a parallelogram, then • ABCD, and • BCDA Theorem of Opposite Sides Mr. Chin-Sung Lin

19. A B 1 3 4 2 D C ERHS Math Geometry Theorem of Opposite Sides Statements Reasons 1. ABCD is a parallelogram 1. Given 2. Connect BD 2. Form two triangles 3. AB || DC and AD || BC 3. Definition of parallelogram 4. 12 and 344. Alternate interior angles 5. BD BD 5. Reflexive property 6. ∆ ABD  ∆ CDB 6. ASA postulate 7. AB  CD and BC  DA 7. CPCTC Mr. Chin-Sung Lin

20. ERHS Math Geometry ABCD is a parallelogram, what’s the perimeter of ABCD ? Application Example 1 A B 15 10 D C Mr. Chin-Sung Lin

21. ERHS Math Geometry ABCD is a parallelogram, what’s the perimeter of ABCD ? perimeter = 50 Application Example 1 A B 15 10 D C Mr. Chin-Sung Lin

22. ERHS Math Geometry ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x Application Example 2 A B x-20 10 D C Mr. Chin-Sung Lin

23. ERHS Math Geometry ABCD is a parallelogram, if the perimeter of ABCD is 80, solve for x x = 50 Application Example 2 A B x-20 10 D C Mr. Chin-Sung Lin

24. A B D C ERHS Math Geometry • Opposite angles of a parallelogram are congruent • If ABCD is a parallelogram, then • AC, and • BD Theorem of Opposite Angles Mr. Chin-Sung Lin

25. A B D C ERHS Math Geometry Theorem of Opposite Angles Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. A and B are supplementary 3. Same side interior angles A and D are supplementary C and B are supplementary 4. AC 4. Supplementary angle theorem BD

26. ERHS Math Geometry ABCD is a parallelogram, what are the values of x and y? Application Example 3 A B 120o 60o y x D C Mr. Chin-Sung Lin

27. ERHS Math Geometry ABCD is a parallelogram, what are the values of x and y? x = 120o y = 60o Application Example 3 A B 120o 60o y x D C Mr. Chin-Sung Lin

28. ERHS Math Geometry ABCD is a parallelogram, what are the values of x and y? Application Example 4 A B X+20 y - 20 180 - y 2x - 60 D C Mr. Chin-Sung Lin

29. ERHS Math Geometry ABCD is a parallelogram, what are the values of x and y? x = 80o y = 100o Application Example 4 A B X+20 y - 20 180 - y 2x - 60 D C Mr. Chin-Sung Lin

30. A B O D C ERHS Math Geometry • The diagonalsof a parallelogram bisect each other • If ABCD is a parallelogram, then • ACandBD bisect each other at O Theorem of Bisecting Diagonals Mr. Chin-Sung Lin

31. 1 3 4 2 A B O D C ERHS Math Geometry Theorem of Bisecting Diagonals Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC 2. Definition of parallelogram 3. 12 and 34 3. Alternate interior angles 4. AB DC 4. Opposite sides congruent 5. ∆ AOB  ∆ COD 5. ASA postulate 6. AO = OC and BO = OD 6. CPCTC 7. AC and BD bisect each other 7. Definition of segment bisector Mr. Chin-Sung Lin

32. ERHS Math Geometry ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ? Application Example 5 A B 6 3 4 O D C Mr. Chin-Sung Lin

33. ERHS Math Geometry ABCD is a parallelogram, if AO = 3, BO = 4 AB = 6, AC + BD = ? AC + BD = 24 Application Example 5 A B 6 3 4 O D C Mr. Chin-Sung Lin

34. ERHS Math Geometry ABCD is a parallelogram, if AO = x+4, BO = 2y-6, CO = 3x-4, an DO = y+2, solve for x and y Application Example 6 A B x+4 2y-6 O y+2 3x-4 D C Mr. Chin-Sung Lin

35. ERHS Math Geometry ABCD is a parallelogram, if AO = x+4, BO = 2y-6, CO = 3x-4, an DO = y+2, solve for x and y x = 4 y = 8 Application Example 6 A B x+4 2y-6 O y+2 3x-4 D C Mr. Chin-Sung Lin

36. A B D C ERHS Math Geometry • The consecutive angles of a parallelogram are supplementary • If ABCD is a parallelogram, then • A and Bare supplementary • C and Dare supplementary • A and Dare supplementary • B and Care supplementary Theorem of Consecutive Angles Mr. Chin-Sung Lin

37. ERHS Math Geometry A B Theorem of Consecutive Angles Statements Reasons 1. ABCD is a parallelogram 1. Given 2. AB || DC and AD || BC 2. Definition of parallelogram 3. AandB,CandD 3. Same-side interior angles AandD,BandC are supplementary are supplementary D C Mr. Chin-Sung Lin

38. ERHS Math Geometry ABCD is a parallelogram, what are the values of x, y and z? Application Example 7 A B 120o x z y D C Mr. Chin-Sung Lin

39. ERHS Math Geometry ABCD is a parallelogram, what are the values of x, y and z? x = 60o y = 120o z = 60o Application Example 7 A B 120o x z y D C Mr. Chin-Sung Lin

40. ERHS Math Geometry ABCD is a parallelogram, what are the values of x and y? Application Example 8 A B X+30 X-30 Y+20 D C Mr. Chin-Sung Lin

41. ERHS Math Geometry ABCD is a parallelogram, what are the values of x and y? x = 90o y = 100o Application Example 8 A B X+30 X-30 Y+20 D C Mr. Chin-Sung Lin

42. ERHS Math Geometry Group Work Mr. Chin-Sung Lin

43. ERHS Math Geometry ABCD is a parallelogram, calculate the perimeter of ABCD Question 1 A B x+30 2y-10 y+10 D C 2x-10 Mr. Chin-Sung Lin

44. ERHS Math Geometry ABCD is a parallelogram, calculate the perimeter of ABCD perimeter = 200 Question 1 A B x+30 2y-10 y+10 D C 2x-10 Mr. Chin-Sung Lin

45. ERHS Math Geometry ABCD is a parallelogram, solve for x Question 2 A B X+30 X-10 O X+10 2X D C Mr. Chin-Sung Lin

46. ERHS Math Geometry ABCD is a parallelogram, solve for x x = 30 Question 2 A B X+30 X-10 O X+10 2X D C Mr. Chin-Sung Lin

47. A X B O D C Y ERHS Math Geometry Given: ABCD is a parallelogram Prove: XO  YO Question 3 Mr. Chin-Sung Lin

48. ERHS Math Geometry Given: ABCD is a parallelogram, BOOD Prove: EOOF Question 4 A E B O D C F Mr. Chin-Sung Lin

49. ERHS Math Geometry Given: ABCD is a parallelogram, AF || CE Prove: FABECD Question 5 A B E F D C Mr. Chin-Sung Lin

50. ERHS Math Geometry Review: Theorems of Parallelogram Theorem of Dividing Diagonals Theorem of Opposite Sides Theorem of Opposite Angles Theorem of Bisecting Diagonals Theorem of Consecutive Angles Mr. Chin-Sung Lin