Parallel Lines and Triangles Skill 18

Objective HSG-CO.10: Students are responsible for using parallel lines to prove theorems about triangles and finding measures of the angles of triangles.

Postulate 11 Parallel Postulate Through a point not on a line, there is one and only one line parallel to the given line. P l

Theorem 16 Triangle Angle-Sum Theorem The sum of the measures of the angles of a triangle is 180. B A C If ∆𝑨𝑩𝑪 Then 𝒎∠𝑨+𝒎∠𝑩+𝒎∠𝑪=𝟏𝟖𝟎

Theorem 16; Triangle Angle-Sum Theorem Given: ∆𝐴𝐵𝐶 Prove: 𝑚∠𝐴+𝑚∠2+𝑚∠𝐶=180 Statement Reason 1) ∆𝐴𝐵𝐶 1) Given 2) Draw 𝑃𝑅 through B, so that 𝑃𝑅 ∥ 𝐴𝐶 . 2) Parallel Postulate 3) 𝑚∠𝑃𝐵𝐶 𝑎𝑛𝑑 𝑚∠3 are supplementary 3) Linear Pair Postulate 4) Def. of Supplementary 4) 𝑚∠𝑃𝐵𝐶+𝑚∠3=180 5) Angle Addition Postulate 5) 𝑚∠𝑃𝐵𝐶=𝑚∠1+𝑚∠2

Theorem 16; Triangle Angle-Sum Theorem Given: ∆𝐴𝐵𝐶 Prove: 𝑚∠𝐴+𝑚∠2+𝑚∠𝐶=180 Statement Reason 6) 𝑚∠1+𝑚∠2+𝑚∠3=180 6) Substitution 7) ∠1≅∠𝐴 and ∠3≅∠𝐶 7) Alt. Int. Angle Theorem 8) 𝑚∠1=𝑚∠𝐴 and 𝑚∠3=𝑚∠𝐶 8) Def. of Congruent Angles 9) 𝑚∠𝐴+𝑚∠2+𝑚∠𝐶=180 9) Substitution Q.E.D.

Theorem 17 B Remote Interior 2 Exterior 1 3 C A Triangle Exterior Angle Theorem The measure of each exterior angle of a triangle equal the sum of the measures of the remote interior angles. B A C 2 3 Remote Interior 1 Exterior If ∆𝑨𝑩𝑪 Then 𝒎∠𝟏=𝒎∠𝟐+𝒎∠𝟑

Definitions An exterior angle of a triangle is an angle formed by a side and an extension of an adjacent side. For each exterior angle of a triangle, the nonadjacent interior angles are its remote interior angles.

#18: Parallel Lines and Triangles Questions? Summarize Notes Homework Video Quiz