1. Building a System of Geometry Knowledge 2.4

2. Algebraic properties

3. Addition Property Subtraction Property If a = b , then a + c = b + c
Homework

4. Multiplication Property
If a = b , then ac = bc
Division Property
If a = b and c ≠ 0 , then a/c = b/c
Homework

5. Substitution Property
If a = b, you may replace a with b in any true equation that has a in it, and the resulting equation will still be true.
Homework

6. Equivalence Properties
Reflexive Property
for any number a, a = a
Symmetric Property
if a = b , b = a
Transitive Property
if a = b and b = c, then a = c
Homework

7. Overlapping SegmentsTheorem
Overlapping segment theorem:
Given a segment with points
A, B, C, and D arranged as shown,
the following statements are true:
If AB = CD then AC = BD
If AC = BD then AB = CD
Homework

8. Overlapping Angles Theorem
Given AED with points B and C in its interior as shown, the following statements are true:
1. If mAEB = mCED, then mAEC = mBED.
2. If mAEC = mBED, then mAEB = mCED.
Homework

9. Equality and Congruency
For all these properties, you can change the equal sign to a congruent sign and they are still true.
In the Reasons column of the proof write Definition of Congruence.
Homework

10. Two Column Proof Homework Statement Reason
Given: AB = CD Prove: AC = BD
Statement
Reason
1. AB = CD
1. Given
2. AB + BC = BC + CD
2. Addition Property
3. AB + BC = AC
3. Segment Addition Postulate
4. BC + CD = BD
4. Segment Addition Postulate
5. AC = BD
5. Substitution Property
Homework

11. Two Column Proof Homework Statement Reason
Given: a = b Prove: a + c = b + c
Statement
Reason
1. a = b
1. Given
2. a + c = a + c
2. Reflexive Property of Equality
3. a + c = b + c
3. Substitution Property of Equality
Homework

12. Two Column Proof Homework Statement Reason
Given: 2x − 5 = Prove: x = 4
Statement
Reason
1. 2x − 5 = 3
1. Given
2. 2x − = 3 + 5
2. Addition Property of Equality
3. 2x = 8
3. Simplify
4. 2x ÷ 2 = 8 ÷ 2
4. Division Property of Equality
5. x = 4
5. Simplify
Homework

13. Two Column Proof Homework Statement Reason Given: B  C Prove: x = 7
2. mB = mC
2. Definition of Congruence
3. 5x + 12 = 47
3. Substitution Property of Equality
4. 5x = 35
4. Subtraction Property of Equality
5. x = 7
5. Division Property of Equality
Homework

14. Two Column Proof Homework Statement Reason Given: m1 + m3 = 180
Prove: 1  2
Statement
Reason
1. m1 + m3 = 180
1. Given
2. m2 + m3 = 180
2. Linear Pair Property
3. m1 + m3 = m2 + m3
3. Substitution Property of Equality
4. m1 = m2
4. Subtraction Property of Equality
5. 1  2
5. Definition of Congruence
Homework

15. Two Column Proof Homework Statement Reason Given: mHGK = mJGL
Prove: 1  3
Statement
Reason
1. mHGK = mJGL
1. Given
2. m∠HGK = m∠1 + m∠2
2. Angle Addition Postulate
3. m∠JGL = m∠3 + m∠2
3. Angle Addition Postulate
4. m∠1 + m∠2 = m∠3 + m∠2
4. Substitution Property of Equality
5. m∠1 = m∠3
5. Subtraction
6. ∠1  ∠3
6. Definition of Congruence
Homework

16. Given: PQ = 2x + 5 QR = 6x – 15 PR = 46 Prove: x = 7
Two Column Proof
Given: PQ = 2x QR = 6x – PR = Prove: x = 7
P Q R
Statement
Reason
1. PQ = 2x + 5, QR = 6x – 15, PR = 46
1. Given
2. PQ + QR = PR
2. Segment Addition Postulate
3. 2x x – 15 = 46
3. Substitution Property of Equality
4. 8x – 10 = 46
4. Simplify
5. 8x = 56
5. Addition Property of Equality
Homework
6. x = 7
6. Division Property of Equality

17. Two Column Proof Homework Statement Reason PR PR
Given: Q is the midpoint of PR. Prove: PQ = and QR = 2 2
Statement
Reason
1. Q is midpoint of PR
1. Given
2. PQ = QR
2. Definition of midpoint
3. PQ + QR = PR
3. Segment Addition Postulate
4. QR + QR = PR & PQ + PQ =PR
4. Substitution Property of Equality
5. 2QR = PR PQ = PR
5. Simplify PR 2
6. Division Property of Equality
Homework
6. QR = PQ =

18. Two Column Proof Homework Statement Reason
Given: 2(3x + 1) = 5x Prove: x = 12
Statement
Reason
1. 2(3x + 1) = 5x + 14
1. Given
2. 6x + 2 = 5x + 14
2. Distributive property
3. x + 2 = 14
3. Subtraction Property of Equality
4. x = 12
4. Subtraction Property of Equality
Homework

19. Two Column Proof Homework Statement Reason
Given: 55z – 3(9z + 12) = – Prove: z = –1
Statement
Reason
1. 55z – 3(9z + 12) = – 64
1. Given
2. 55z – 27z – 36 = – 64
2. Distributive Property
3. 28z – 36 = – 64
3. Simplify
4. 28z = –28
4. Addition Property of Equality
5. z= – 1
5. Division Property of Equality
Homework

20. Summary of Properties
Homework

21. Assignment
Geometry:
2.4A and 2.4B
Section