Angle Relationships in Triangles

Angle Relationships in Triangles Goal: To learn angle properties. Vertex point where two or more lines meet. Interior angle angle formed on the inside of a polygon by two sides meeting at a vertex. Exterior angle angle formed on the outside of a geometric shape by extendi ng one of the lines past the vertex. The sum of the interior angles of a triangle is 180 ◦ ∠𝐴𝐵𝐶 + ∠𝐵𝐶𝐴 + ∠𝐶𝐴𝐵 = 180 ◦ G.Sketchpad 𝑚 ∠𝐴𝐵𝐶 = 70. 70 ◦ 𝑚 ∠𝐵𝐶𝐴 = 70. 70 ◦ 𝑚 ∠𝐶𝐴𝐵 = 70. 70 ◦ 𝑚 ∠𝐴𝐵𝐶 + 𝑚 ∠𝐵𝐶𝐴 + 𝑚 ∠𝐶𝐴𝐵 = 180. 00 The sum of the exterior angles of a triangle is 360 ◦ Note that in angle names with three letters, the middle letter is always the vertex of the angle. Angle Properties Complementary Angles

Supplementary Angles Vertically Opposite Angles Exterior Angle of a Triangle Transversal Parallel Lines

Alternate Angles are Equal Corresponding Angles are Equal

Co-interior Angles Example: a) Find the measure of the exterior angle at vertex X . Explain your reasoning. I: 𝑋 + 120 ◦ + 140 ◦ = 360 ◦ 𝑋 + 260 ◦ = 360 ◦ 𝑋 = 360 − 260 𝑋 = 100 Another way, II: 𝑎 + 120 ◦ = 180 ◦ 𝑎 = 180 ◦ − 120 ◦ 𝑎 = 60 ◦ 𝑏 + 140 ◦ = 180 ◦ 𝑏 = 180 ◦ − 140 ◦ 𝑏 = 40 ◦

So, the exterior angle at vertex X is 𝑎 + 𝑏 = 40 ◦ + 60 ◦ = 100 ◦ b) Helena says this exterior angle measures . 110 ◦ Is she right? Explain She is right since 40 ◦ + 70 ◦ = 110 ◦ (sum of the opposite interior angles is 110 𝑜 Another way: 40 ◦ + 70 ◦ + 𝑦 = 180 ◦ 𝑦 = 180 − 110 𝑦 = 70 Since x and y are supplementary angles then 𝑥 + 𝑦 = 180 ◦ 𝑥 + 70 ◦ = 180 ◦ 𝑥 = 180 − 70 𝑥 = 110