Proving Lines Parallel Skill 16

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

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

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

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.

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

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

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.

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

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.

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+9 +111=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.

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.

#16: Proving Lines Parallel Questions? Summarize Notes Homework Worksheet Quiz