SECTION 2-7 Proving Segment Relationships
SECTION 2-7Proving 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, then AB ≅ 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 propertyof 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 propertyof 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 propertyof Equality
4. AB ≅ EF Def. of ≅ segments
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.
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
CD
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
CD
Given: AB = AD, AB ≅ BC , and BC ≅ CD
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
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
PROBLEM SET
PROBLEM SET
p. 145 #1-13, 15, 17, 18
“Trust yourself. Think for yourself. Act for yourself. Speak for yourself. Be yourself. Imitation is suicide.” - Marva Collins