1. Proving Lines Parallel
Skill 16

2. Objective
HSG-CO.9: Students are responsible for proving theorems about parallel lines, and determining whether two lines are parallel.

3. Theorem 9
Converse to Corresponding Angles Theorem
If two lines an a transversal form corresponding angles that are congruent, then the lines are parallel. t 4 3 6 7 5 8 2 1 l
If ∠𝟐≅∠𝟔
Then 𝒍∥𝒎 m

4. Theorem 10 t 4 3 6 7 5 8 2 1 l m Converse to Alt. Int. Angles Theorem
If two lines an a transversal form alternate interior angles that are congruent, then the lines are parallel. t 4 3 6 7 5 8 2 1 l
If ∠𝟒≅∠𝟔
Then 𝒍∥𝒎 m

5. Theorem 10; Converse to AIA Theorem
Given: ∠4≅∠6 Prove: 𝒍∥𝒎
Statement Reason
1) ∠4≅∠6
1) Given
2) Vertical Angles Theorem
2) ∠2≅∠4
3) ∠2≅∠6
3) Alg. (Transitive)
4) 𝑙∥𝑚
4) Converse to Corr. Angles Theorem
Q.E.D.

6. Theorem 11
Converse to Same-Side Int. Angles Postulate
If two lines an a transversal form same-side interior angles that are supplementary, then the lines are parallel. t 4 3 6 7 5 8 2 1 l
If ∠𝟑≅∠𝟔
Then 𝒍∥𝒎 m

7. Theorem 12 t 4 3 6 7 5 8 2 1 l m Converse to Alt. Ext. Angles Theorem
If two lines an a transversal form alternate exterior angles that are congruent, then the lines are parallel. t 4 3 6 7 5 8 2 1 l
If ∠𝟏≅∠𝟕
Then 𝒍∥𝒎 m

8. Theorem 12; Converse to AIA Theorem
Given: ∠1≅∠7 Prove: 𝒍∥𝒎
Statement Reason
1) ∠1≅∠7
1) Given
2) Vertical Angles Theorem
2) ∠1≅∠3
3) ∠3≅∠7
3) Alg. (Transitive)
4) 𝑙∥𝑚
4) Converse to Corr. Angles Theorem
Q.E.D.

9. Example 1; Identify Parallel Lines
Which lines are parallel if ∠1≅∠2?
Justify your answer. 1 2 3 4 5 6 7 8 m l a b
If ∠𝟏≅∠𝟐
Then 𝒂∥𝒃
By Converse to Corresponding Angles Theorem

10. Example 2; Identify Parallel Lines
Given: 𝑚∠3+𝑚∠6=180 Prove: 𝒍∥𝒎
Statement Reason
1) Given
1) 𝑚∠3+𝑚∠6=180
2) Linear Pair Post./ Def. of Supp. ∠’s
2) 𝑚∠6+𝑚∠7=180
3) 𝑚∠3+𝑚∠6=𝑚∠6+𝑚∠7
3) Transitive Prop.
4) 𝑚∠3=𝑚∠7
4) Subtraction
5) Def. of ≅ ∠’s
5) ∠3≅∠7 3 1 5 6 7 m l
6) Converse to Corr. Angles Theorem
6) 𝑙∥𝑚
Q.E.D.

11. Example 3; Prove the value of x
Given: 𝑎∥𝑏, ∠1= 2𝑥+9 °, and ∠2=111°
Prove: 𝑥=30
Statement Reason
1) Given
1)𝑎∥𝑏, ∠1= 2𝑥+9 °, & ∠2=111°
2) S.S.I.A Postulate
2) 𝑚∠1+𝑚∠2=180
3) 2𝑥 =180
3) Substitution Prop.
4) 2𝑥+120=180
4) Combine Terms
5) 2𝑥=60
5) Subtraction 1 2 b a
6) 𝑥=30
6) Division
Q.E.D.

12. Example 3; Prove the value of x
Given: 𝑎∥𝑏, ∠1=55°, and ∠2= 3𝑥−2 °
Prove: 𝑥=19
Statement Reason
1) Given
1) 𝑎∥𝑏, ∠1=55°, & ∠2= 3𝑥−2 °
2) ∠1≅∠2
2) Corr. ∠’s Thm.
3) 𝑚∠1=𝑚∠2
3) Def. of ≅ ∠’s
4) 55=3𝑥−2
4) Substitution 1 2 b a
5) 3𝑥=57
5) Addition
6) 𝑥=19
6) Division
Q.E.D.

13. #16: Proving Lines Parallel
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