Section 3.5 Parallel Lines and Triangles

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

Solve it:

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

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

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.

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.

Proofs Using Parallel Lines Proof of Triangle Angle-Sum Theorem

Proofs Using Parallel Lines

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

Parallel Lines and Triangles What are the values of x and y in the diagram. Statement Reason 59 + 43 + 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.

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.

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

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?

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

Lesson Check

Lesson Check

Some More Practice