Math 2 Geometry Based on Elementary Geometry, 3rd ed, by Alexander

1
Math 2 GeometryBased on Elementary Geometry,
3rd ed, by Alexander Koeberlein

• 1.5
• Introduction to Geometric Proof

2
Properties of Equality

• Addition property of equality
• If a b, then a c b c
• Subtraction property of equality
• If a b, then a c b c
• Multiplication property of equality
• If a b, then ac bc
• Division property of equality
• If a b and c ? 0, then a/c b/c

3
Properties of Inequality

• Addition property of inequality
• If a gt b, then a c gt b c
• Subtraction property of inequality
• If a gt b, then a c gt b c
• Multiplication property of inequality
• If a gt b, and c gt 0, then ac gt bc
• Division property of inequality
• If a gt b and c gt 0, then a/c gt b/c

4
More Algebra Properties

• Distributive property
• a(b c) ab ac
• Substitution property
• If a b, then a replaces b in any equation
• Transitive property
• If a b and b c, then a c

5
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2. 3. 4. 5. x 6 Given 2. 3. 4. 5.
6
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2x 6 4 10 3. 4. 5. x 6 Given 2. 3. 4. 5.
7
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2x 6 4 10 3. 4. 5. x 6 Given Distributive Property 3. 4. 5.
8
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2x 6 4 10 2x 2 10 4. 5. x 6 Given Distributive Property 3. 4. 5.
9
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2x 6 4 10 2x 2 10 4. 5. x 6 Given Distributive Property Substitution 4. 5.
10
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2x 6 4 10 2x 2 10 2x 12 x 6 Given Distributive Property Substitution 4. 5.
11
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2x 6 4 10 2x 2 10 2x 12 x 6 Given Distributive Property Substitution Add. Prop. of Equality 5.
12
Given 2(x 3) 4 10 Prove x 6

• Proof

Statements Reasons
2(x 3) 4 10 2x 6 4 10 2x 2 10 2x 12 x 6 Given Distributive Property Substitution Add. Prop. of Equality 5. Division Prop. of Eq.
13
Given A-P-B on seg AB Prove AP AB - PB

• Proof

Statements Reasons
1. A-P-B on seg AB 2. ?. AP AB - PB 1. Given 2. ?.
14
Given A-P-B on seg AB Prove AP AB - PB

• Proof

Statements Reasons
1. A-P-B on seg AB 2. ?. AP AB - PB 1. Given 2. Segment-Addition Postulate ?.
15
Given A-P-B on seg AB Prove AP AB - PB

• Proof

Statements Reasons
1. A-P-B on seg AB 2. AP PB AB ?. AP AB PB 1. Given 2. Segment-Addition Postulate ?.
16
Given A-P-B on seg AB Prove AP AB - PB

• Proof

Statements Reasons
1. A-P-B on seg AB 2. AP PB AB 3. AP AB PB 1. Given 2. Segment-Addition Postulate 3.
17
Given A-P-B on seg AB Prove AP AB - PB

• Proof

Statements Reasons
1. A-P-B on seg AB 2. AP PB AB 3. AP AB PB 1. Given 2. Segment-Addition Postulate 3. Subtr. Prop. of Equality
18
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons

19
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. ?. MP gt NQ 1. Given 2. ?.
20
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. ?. MP gt NQ 1. Given 2. Addn prop of Inequality ?.
21
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. MN NP gt PQ NP ?. MP gt NQ 1. Given 2. Addn prop of Inequality ?.
22
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. MN NP gt PQ NP 3. MN NP gt NP PQ ?. MP gt NQ 1. Given 2. Addn prop of Inequality 3. ?.
23
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. MN NP gt PQ NP 3. MN NP gt NP PQ ?. MP gt NQ 1. Given 2. Addn prop of Inequality 3. Commutative prop. of addn ?.
24
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. MN NP gt PQ NP 3. MN NP gt NP PQ 4. MN NP MP and NP PQ NQ ?. MP gt NQ 1. Given 2. Addn prop of Inequality 3. Commutative prop. of addn 4. ?.
25
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. MN NP gt PQ NP 3. MN NP gt NP PQ 4. MN NP MP and NP PQ NQ ?. MP gt NQ 1. Given 2. Addn prop of Inequality 3. Commutative prop. of addn 4. Segment Addition Postulate ?.
26
Given MN gt PQ Prove MP gt NQ

• Proof

Statements Reasons
1. MN gt PQ 2. MN NP gt PQ NP 3. MN NP gt NP PQ 4. MN NP MP and NP PQ NQ 5. MP gt NQ 1. Given 2. Addn prop of Inequality 3. Commutative prop. of addn 4. Segment Addition Postulate 5. Substitution