3.5 Parallel Lines and Triangles

Objectives To use parallel lines to prove a theorem about triangles To find measures of angles in triangles

Not Parallel Parallel Postulate Postulate 3.3 l P Through a point not on a line, there is one and only one line parallel to the given line Not Parallel l P

Triangle Sum Theorem 𝑚∠𝐴+𝑚∠𝐵+𝑚∠𝐶=180 Theorem 3.10 – the sum of the measures of the angles of a triangle is 180 𝑚∠𝐴+𝑚∠𝐵+𝑚∠𝐶=180

Proof of Triangle Sum Theorem 𝑚∠1+𝑚∠2+𝑚∠3=180 2 1 3 3 1 2 1 3 Alternate Interior Angles

Example Check Work 25+65+90=180 90+2𝑥+1+5𝑥+5=180 96+7𝑥=180 2 12 +1=𝟐𝟓 Solve for the angles of the triangle. 90+2𝑥+1+5𝑥+5=180 96+7𝑥=180 2 12 +1=𝟐𝟓 7𝑥=84 𝑥=12 5 12 +5=𝟔𝟓

Example Check Work 62+71+47=180 7 9 +8=𝟕𝟏 7𝑥−1+7𝑥+8+5𝑥+2=180 19𝑥+9=180 Solve for the angles of the triangle 7 9 +8=𝟕𝟏 7𝑥−1+7𝑥+8+5𝑥+2=180 19𝑥+9=180 19𝑥=171 𝑥=9 5 9 +2=𝟒𝟕 7 9 −1=𝟔𝟐

Exterior Angles of Triangle Exterior Angles – angle formed by a side and an extension of an adjacent side straight angle = 180 3 1 <1, <2, and <3 are Exterior Angles 2 The Exterior angles are not the straight angles!!!

Remote Interior Angles Remote Interior angles – the two nonadjacent angles to an exterior angle 2 3 1 4 6 5 Exterior Angle Remote Interior Angles <1 <4 and <5 <2 <4 and <6 <3 <5 and <6

Triangle Exterior Angle Theorem Theorem 3.11 – the measure of each exterior angle of a triangle equals the sum of the measures of its two remote interior angles 1 4 5 𝑚∠1=𝑚∠4+𝑚∠5

Example What is the value of x? 2𝑥+3𝑥=100 5𝑥=100 𝑥=20

Example Find the measure of each angle. 𝑥+60=2𝑥+4 𝑥+56=2𝑥 56=𝑥

Example 4𝑥+2+2𝑥−9=5𝑥+13 6𝑥−7=5𝑥+13 6𝑥=5𝑥+20 𝑥=20 What is the value of x? 4𝑥+2+2𝑥−9=5𝑥+13 6𝑥−7=5𝑥+13 6𝑥=5𝑥+20 𝑥=20

Take Home Message Interior Angles of a Triangle  Sum is 180 Exterior Angles of a Triangle  Exterior Angles EQUALS the sum of the two remote interior angles