1. 3.3 Proofs with parallel lines

2. What we will learn Use corresponding angles converse
Prove lines parallel
Use transitive property of parallel lines

3. Ex. 1 using theorems to prove parallel
Find value of x that makes 𝑚∥𝑛
Corresponding angles congruent thm
3𝑥+5=65
−5 −5
3𝑥=60
3𝑥 3 = 60 3
𝑥=20
3x+5 m 65 n

4. Your Practice Find value of x that makes 𝑚∥𝑛 3𝑥−15+150=180 3𝑥+135=180
−135 −135
3𝑥=45
3𝑥 3 = 45 3
𝑥=15
150
3x-15

5. Ex. 2 proving lines parallel
Statement
Reason
Given: ∠1 𝑎𝑛𝑑 ∠3 are supplementary
Prove: 𝑚∥𝑛
1. ∠𝟏 𝒂𝒏𝒅 ∠𝟑 are supplementary
1. Given
2. ∠𝟏≅∠𝟐
2. Vertical angles
3. ∠𝟐 𝒂𝒏𝒅 ∠𝟑 are supplementary
3. substitution 1 m 2
4. 𝒎∥𝒏
4. Thm 3.8 3 n

6. Your practice Given: ∠1≅∠2, ∠3≅∠4 Prove: 𝐴𝐵 ∥ 𝐶𝐷 Statement Reason
1. ∠1≅∠2, ∠3≅∠4
1. Given
2. ∠2≅∠3
2. Vert. Angles
3. ∠1≅∠3
3. Trans. Prop
4. ∠1≅∠3
4. Trans. Prop
5. 𝐴𝐵 ∥ 𝐶𝐷
5. Thm 3.6 A D 1 E 2 3 4 B C

7. Ex. 3 is there enough information
Given: 𝑟∥𝑠 𝑎𝑛𝑑 ∠1≅∠3
Can you prove 𝑝∥𝑞?
Yes, because ∠1≅∠2 by corresponding angles. ∠2≅∠3 by substitution. Therefore 𝑝∥𝑞 by Thm 3 p 2 1 q r s

8. Ex. 4 transitive property of parallel lines
Find 𝑚∠8
115+𝑚∠8=180
− −115
𝑚∠8=65