1. Section 3.5 Parallel Lines and Triangles

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

3. Solve it:

4. Solve it Solution:
In the Solve It, you may have discovered that you can rearrange the corners of the triangles to form a straight angle. You can do this for any triangle.
1800
<1 + <2 + <3 = 180

5. Parallel Lines
The sum of the angle measures of a triangle is always the same. We will use parallel lines to prove this theorem.

6. Parallel Lines and Triangle
The sum of the angle measures of a triangle is always the same. We will use parallel lines to prove this theorem.

7. Parallel Lines and Triangles
To prove the Triangle Angle-Sum Theorem, we must use an auxiliary line. An auxiliary line is a line that you add to the diagram to help explain relationships in proofs. The red line is the diagram is an auxiliary line.

8. Proofs Using Parallel Lines
Proof of Triangle Angle-Sum Theorem

9. Proofs Using Parallel Lines

10. Parallel Lines and Triangles
What are the values of x and y in the diagram.

11. Parallel Lines and Triangles
What are the values of x and y in the diagram.
Statement
Reason
x = 180
Triangle Angle Sum
102 + x = 180
Like Terms / Addition
x = 78
Subtraction
m< ADB + m<CDB = 180
Linear Pair
x + y = 180
Substitution
78 + y = 180
Substitution
y = 180
Subtraction.

12. Exterior Angle of a Polygon
An exterior angle of a polygon is an angle formed by a side and an extension of an adjacent side. For each exterior angle of a triangle, the two nonadjacent angles are its remote interior angles. In the triangle below <1 is an exterior angle and <2 and <3 are its remote interior angles.

13. Exterior Angle
The theorem below states the relationship between an exterior angle and its two remote interior angles. + =

14. Real World
When radar tracks an object, the reflection of signals off the ground can result in clutter. Clutter causes the receiver to confuse the real object with its reflection, called a ghost. At the right, there is a radar receiver at A, an airplane at B, and the airplane’s ghost at D. What is the value of x?

15. Real World + = Exterior Angle x + 30 = 80 Substitution X = 50
m<A +m<B = m<BCD
x + 30 = 80
Substitution
X = 50
Subtraction

16. Lesson Check

17. Lesson Check

18. Some More Practice