Building a System of
Geometry Knowledge 2.4
Algebraic properties
Addition Property

If a = b , then a + c = b + c
Subtraction Property

If a = b , then a – c = b – c
Multiplication Property

If a = b , then ac = bc
Division Property

If a = b and c ≠ 0 , then
a/c = b/c
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.
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

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
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.
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.
Two Column Proof
Given: AB = CD
Statement
Prove: AC = BD
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
Two Column Proof
Given: a = b
Statement
Prove: a + c = b + c
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
Two Column Proof
Given: 2x − 5 = 3
Statement
Prove: x = 4
Reason
1. 2x − 5 = 3
1. Given
2. 2x − 5 + 5 = 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
Two Column Proof
Given: B  C
Statement
Prove: x = 7
Reason
1. B  C
1. Given
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
Two Column Proof
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
Two Column Proof
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
Two Column Proof
Given: PQ = 2x + 5
QR = 6x – 15
Statement
PR = 46
Prove: x = 7
Reason
1. PQ = 2x + 5, QR = 6x – 15, PR = 46
1. Given
2. PQ + QR = PR
2. Segment Addition Postulate
3. 2x + 5 + 6x – 15 = 46
3. Substitution Property of Equality
4. 8x – 10 = 46
4. Simplify
5. 8x = 56
5. Addition Property of Equality
6. x = 7
6. Division Property of Equality
Two Column Proof
PR
PR
Given: Q is the midpoint of PR. Prove: PQ = 2 and QR = 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
5. Simplify
PR
6. QR =
2
2PQ = PR
PR
PQ =
2
6. Division Property of Equality
Two Column Proof
Given: 2(3x + 1) = 5x + 14
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
Two Column Proof
Given: 55z – 3(9z + 12) = –64.
Statement
Prove: z = –1
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
Summary of Properties
Assignment
Geometry:
2.4A and 2.4B
Section 9 - 24