1. 3.5 Parallel Lines and Triangles

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

3. 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

4. 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

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

6. 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
=𝟐𝟓
7𝑥=84
𝑥=12
=𝟔𝟓

7. 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

8. 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!!!

9. 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

10. 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

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

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

13. 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

14. 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