Transcript Lesson 3.2 Proofs

3.2 Proving Lines Parallel
Proofs
Given:∠1 𝑎𝑛𝑑 ∠4 are supplementary.
Prove: a∥b
1.
2.
3.
4.
Statements
∠1 𝑎𝑛𝑑 ∠4 are
supplementary.
∠1 ≅ ∠2 and ∠3 ≅ ∠4
∠2 𝑎𝑛𝑑 ∠3 are
supplementary
a∥b
Reasons
1. Given
2. Vertical Angles Thm.
3. Substitution Property
4. Converse of Same – Side
Interior Angles
Given: ∠1 ≅ ∠4
Prove: ∠2 ≅ ∠3
1.
2.
3.
4.
5.
Statements
∠1 ≅ ∠4
∠1 ≅ ∠2
∠2 ≅ ∠4
∠3 ≅ ∠4
∠2 ≅ ∠3
Reasons
1.
2.
3.
4.
5.
Given
Vertical Angles Thm.
Transitive Prop.
Vertical Angles Thm.
Transitive Prop.
Given: g∥h and ∠1 ≅ ∠2
Prove: p∥r
Statements
1. g∥h
2. ∠1 ≅ ∠3
3. ∠1 ≅ ∠2
4. ∠2 ≅ ∠3
5. p∥r
Reasons
1.
2.
3.
4.
5.
Given
Corresponding Angles Thm.
Given
Transitive Prop.
Converse Corresponding
Angles.
Given: q∥r
Prove: ∠1 ≅ ∠3
Statements
1. q∥r
2. ∠1 ≅ ∠2
3. ∠2 ≅ ∠3
4. ∠1 ≅ ∠3
Reasons
1. Given
2. Vertical Angles
3. Corresponding Angles Thm.
4. Transitive Property
Given: a∥b, c∥d
Prove: ∠1 and ∠4 are
supplementary
1.
2.
3.
4.
5.
6.
Statements
1.
∠1 ≅ ∠2
2.
c∥d
∠2 and ∠3 are supplementary 3.
4.
a∥b
5.
∠3 ≅ ∠4
∠1 and ∠4 are supplementary 6.
Reasons
Vertical Angles Thm.
Given
Same Side Interior Thm.
Given
Alt. Int. Angles Thm
Substitution
Given: a∥c, b∥c, m∠2 =65
Prove: m∠1 =65
1.
2.
3.
4.
Statements
a∥c, b∥c, m ∠2=65
a∥b
∠1 ≅ ∠2
m∠1 =65
Reasons
1.
2.
3.
4.
Given
Transitive Prop.
Alt. Int. Angles Thm.
Def. of Congruent