Proving Segment Relationships

1. SECTION 2-7
Proving Segment Relationships

2. ESSENTIAL QUESTIONS
How do you write proofs involving segment addition?
How do you write proofs involving segment
congruence?

3. POSTULATES & THEOREMS
Ruler Postulate:
Segment Addition Postulate:

4. POSTULATES & THEOREMS
Rule r P o s t u l a t e : The points on any line or segment can be put
into one-to-one correspondence with real numbers
Segment Addition Postulate:

5. POSTULATES & THEOREMS
Rule r P o s t u l a t e : The points on any line or segment can be put
into one-to-one correspondence with real numbers
You can measure the distance between two points
Segment Addition Postulate:

6. POSTULATES & THEOREMS
Rule r P o s t u l a t e : The points on any line or segment can be put
into one-to-one correspondence with real numbers
You can measure the distance between two points
Segm e n t A d d i t i o n P o s t u l a t e : If A, B, and C are collinear, then B is
between A and C if and only if (IFF) AB + BC = AC

7. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence:
Transitive Property of Congruence:

8. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence:
Transitive Property of Congruence:

9. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence: If AB ≅ CD, then CD ≅ AB
Transitive Property of Congruence:

10. THEOREM 2.2 - PROPERTIES OF
SEGMENT CONGRUENCE
Reflexive Property of Congruence:
Symmetric Property of Congruence: If AB ≅ CD, then CD ≅ AB
Transitive Property of Congruence: If AB ≅ CD and CD ≅ EF, then
AB ≅ EF

11. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD

12. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD

13. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given

14. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD

15. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments

16. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC

17. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality

18. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC

19. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC Segment Addition

20. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC Segment Addition
5. CD + BC = AC

21. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
1. AB ≅ CD Given
2. AB = CD Def. of ≅ segments
3. BC = BC Reflexive property of equality
4. AB + BC = AC Segment Addition
5. CD + BC = AC Substitution prop. of equality

22. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD

23. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
6. CD + BC = BD

24. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
6. CD + BC = BD Segment Addition

25. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
6. CD + BC = BD Segment Addition
7. AC = BD

26. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
6. CD + BC = BD Segment Addition
7. AC = BD Substitution property of equality

27. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
6. CD + BC = BD Segment Addition
7. AC = BD Substitution property of equality
8. AC ≅ BD

28. EXAMPLE 1
Prove that if AB ≅ CD, then AC ≅ BD
6. CD + BC = BD Segment Addition
7. AC = BD Substitution property of equality
8. AC ≅ BD Def. of ≅ segments

29. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF

30. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF

31. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given

32. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF

33. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments

34. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF

35. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF Transitive property
of Equality

36. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF Transitive property
of Equality
4. AB ≅ EF

37. PROOF
The Transitive Property of Congruence
If AB ≅ CD and CD ≅ EF, then AB ≅ EF
1. AB ≅ CD and CD ≅ EF Given
2. AB = CD and CD = EF Def. of ≅ segments
3. AB = EF Transitive property
of Equality
4. AB ≅ EF Def. of ≅ segments

38. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.

39. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.
A B
D C

40. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.
A B
D C
Given: AB = AD, AB ≅ BC, and BC ≅ CD

41. EXAMPLE 2
Matt Mitarnowski is designing a badge for his club. The length of
the top edge of the badge is equal to the length of the left edge of the
badge. The top edge of the badge is congruent to the right edge of the
badge, and the right edge of the badge is congruent to the bottom edge
of the badge. Prove that the bottom edge of the badge is congruent to
the left edge of the badge.
A B
D C
Given: AB = AD, AB ≅ BC, and BC ≅ CD
Prove: AD ≅ CD

42. EXAMPLE 2 A B
D C

43. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C

44. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C
Given

45. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD
A B
D C

46. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C
Given
2. AB ≅ AD Def. of ≅ segments

47. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD
A B
D C

48. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property

49. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB

50. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB Symmetric property

51. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB Symmetric property
5. AD ≅ CD

52. EXAMPLE 2
1. AB = AD,
AB ≅ BC, and
BC ≅ CD
A B
D C
Given
2. AB ≅ AD Def. of ≅ segments
3. AB ≅ CD Transitive property
4. AD ≅ AB Symmetric property
5. AD ≅ CD Transitive property

53. PROBLEM SET

54. PROBLEM SET
p. 145 #1-13, 15, 17, 18
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