Building a System of Building a System of Geometry Knowledge 2.4Geometry Knowledge 2.4

Algebraic propertiesAlgebraic properties

Addition PropertyAddition Property If a = b , then a + c = b + cIf a = b , then a + c = b + c

Subtraction PropertySubtraction Property If a = b , then a – c = b – cIf a = b , then a – c = b – c

Multiplication PropertyMultiplication Property If a = b , then ac = bcIf a = b , then ac = bc

Division PropertyDivision Property If a = b and c If a = b and c ≠ 0 , then≠ 0 , then a/c = b/ca/c = b/c

Substitution PropertySubstitution Property

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

Equivalence PropertiesEquivalence Properties Reflexive PropertyReflexive Property

for any number a, a = afor any number a, a = a Symmetric PropertySymmetric Property

if a = b , b = aif a = b , b = a Transitive PropertyTransitive Property

if a = b and b = c, then a = if a = b and b = c, then a = cc

Overlapping SegmentsTheoremOverlapping SegmentsTheorem

Overlapping segment theorem:Given a segment with pointsA, 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 AnglesOverlapping AnglesTheoremTheorem

Given AED with points B and C in its interior as shown, the following statements are true:

1. If mAEB = mCED, then mAEC = mBED.

2. If mAEC = mBED, then mAEB = mCED.

Equality and CongruencyEquality and Congruency

For all these properties, you can For all these properties, you can change the equal sign to a congruent change the equal sign to a congruent sign and they are still true. sign and they are still true.

In the Reasons column of the proof In the Reasons column of the proof write Definition of Congruence.write Definition of Congruence.

Two Column ProofTwo Column Proof

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

Given: AB = CD Prove: AC = BD

Two Column ProofTwo Column ProofGiven: 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

Two Column ProofTwo Column Proof

Statement 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

Given: 2x − 5 = 3 Prove: x = 4

Two Column ProofTwo Column Proof

Statement Reason

1. B C 1. Given

2. mB = mC 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

Given: B C Prove: x = 7

Two Column ProofTwo Column Proof

Statement Reason

1. m1 + m3 = 180 1. Given

2. m2 + m3 = 180 2. Linear Pair Property

3. m1 + m3 = m2 + m3 3. Substitution Property of Equality

4. m1 = m2 4. Subtraction Property of Equality

5. 1 2 5. Definition of Congruence

Given: m1 + m3 = 180

Prove: 1 2

Two Column ProofTwo Column Proof

Statement Reason

1. mHGK = mJGL 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

Given: mHGK = mJGL

Prove: 1 3

5. m 1 = m 3∠ ∠ 5. Subtraction

6. 1 ∠ 3∠ 6. Definition of Congruence

Given: PQ = 2x + 5 QR = 6x – 15 PR = 46 : PQ = 2x + 5 QR = 6x – 15 PR = 46 Prove: x = 7: x = 7 P

Q

RTwo Column ProofTwo Column Proof

Statement 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 ProofTwo Column Proof

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 2PQ = PR 5. Simplify

Given: Q is the midpoint of PR. Prove: PQ = and QR = Given: Q is the midpoint of PR. Prove: PQ = and QR =

6. QR = PQ = 6. Division Property of Equality

PRPRPRPR22 22

PR2

PR2

Two Column ProofTwo Column Proof

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

Given: 2(3x + 1) = 5x + 14 Prove: x = 12Given: 2(3x + 1) = 5x + 14 Prove: x = 12

Two Column ProofTwo Column Proof

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

Given: 55z Given: 55z – 3(9z + 12) = 3(9z + 12) = –64. Prove: z = 64. Prove: z = –1 1

Summary of PropertiesSummary of Properties

AssignmentAssignmentGeometry:Geometry:

2.4A and 2.4B2.4A and 2.4B

Section 9 - 24Section 9 - 24