SECTION 2-5Proving Segment Relationships

ESSENTIAL QUESTIONS

How do you write proofs involving segment addition?

How do you write proofs involving segment congruence?

POSTULATES & THEOREMS

Ruler Postulate:

Segment Addition Postulate:

POSTULATES & THEOREMS

Ruler Postulate: The points on any line or segment can be put into one-to-one correspondence with real numbers

Segment Addition Postulate:

POSTULATES & THEOREMS

Ruler Postulate: The points on any line or segment can be put into one-to-one correspondence with real numbersYou can measure the distance between two points

Segment Addition Postulate:

POSTULATES & THEOREMS

Ruler Postulate: The points on any line or segment can be put into one-to-one correspondence with real numbersYou can measure the distance between two points

Segment Addition Postulate: If A, B, and C are collinear, then B is between A and C if and only if (IFF) AB + BC = AC

THEOREM 2.2 - PROPERTIES OF SEGMENT CONGRUENCE

Reflexive Property of Congruence:

Symmetric Property of Congruence:

Transitive Property of Congruence:

THEOREM 2.2 - PROPERTIES OF SEGMENT CONGRUENCE

Reflexive Property of Congruence:

Symmetric Property of Congruence:

Transitive Property of Congruence:

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:

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, thenAB ≅ EF

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

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

1. AB ≅ CD

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

1. AB ≅ CD Given

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

1. AB ≅ CD Given

2. AB = CD

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

1. AB ≅ CD Given

2. AB = CD Def. of ≅ segments

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

1. AB ≅ CD Given

2. AB = CD Def. of ≅ segments

3. BC = BC

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

1. AB ≅ CD Given

2. AB = CD Def. of ≅ segments

3. BC = BC Reflexive property of equality

EXAMPLE 1Prove 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

EXAMPLE 1Prove 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

EXAMPLE 1Prove 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

EXAMPLE 1Prove 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

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

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

6. CD + BC = BD

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

6. CD + BC = BD Segment Addition

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

7. AC = BD

6. CD + BC = BD Segment Addition

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

7. AC = BD Substitution property of equality

6. CD + BC = BD Segment Addition

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

7. AC = BD Substitution property of equality

6. CD + BC = BD Segment Addition

8. AC ≅ BD

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

7. AC = BD Substitution property of equality

6. CD + BC = BD Segment Addition

8. AC ≅ BD Def. of ≅ segments

PROOFThe Transitive Property of Congruence

If AB ≅CD and CD ≅ EF, then AB ≅ EF

PROOFThe Transitive Property of Congruence

If AB ≅CD and CD ≅ EF, then AB ≅ EF

1. AB ≅CD and CD ≅ EF

PROOFThe Transitive Property of Congruence

If AB ≅CD and CD ≅ EF, then AB ≅ EF

1. AB ≅CD and CD ≅ EF Given

PROOFThe 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

PROOFThe 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

PROOFThe 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

PROOFThe 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

PROOFThe 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

PROOFThe 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

EXAMPLE 2Matt 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.

EXAMPLE 2Matt 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

CD

EXAMPLE 2Matt 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

CD

Given: AB = AD, AB ≅ BC, and BC ≅CD

EXAMPLE 2Matt 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

CD

Prove: AD ≅CD

Given: AB = AD, AB ≅ BC, and BC ≅CD

EXAMPLE 2 A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CD

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD Def. of ≅ segments

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD Def. of ≅ segments

3. AB ≅CD

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD Def. of ≅ segments

3. AB ≅CD Transitive property

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD Def. of ≅ segments

3. AB ≅CD Transitive property

4. AD ≅ AB

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD Def. of ≅ segments

3. AB ≅CD Transitive property

4. AD ≅ AB Symmetric property

A B

CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD Def. of ≅ segments

3. AB ≅CD Transitive property

4. AD ≅ AB Symmetric property

A B

CD

5. AD ≅CD

EXAMPLE 2

1. AB = AD,AB ≅ BC, and

BC ≅CDGiven

2. AB ≅ AD Def. of ≅ segments

3. AB ≅CD Transitive property

4. AD ≅ AB Symmetric property

A B

CD

5. AD ≅CD Transitive property