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Home > GC > Chapter 6 > Lesson 6.2.5 > Problem 6-84

6-84.

For each part below, decide if the triangles are similar. If they are similar, use their similarity to solve for . If they are not similar, explain why not.

  1. A triangle with an internal horizontal line drawn parallel to the base forming an internal triangle sharing the top vertex. For the internal triangle, the left side is 8 and the right side is 13. For the main triangle the left side is 8 + 5 and the right side s 13 + x.

  1. A transversal segment connects the ends of two parallel line segments, 5, and x. A triangle is formed when the opposite end of the parallel segment, 5, has a diagonal line segment, 8, that connects at the point where the transversal connects to the parallel segment, x. And at the end of the parallel segment, x, another diagonal line, 4, connects to a point further along on the transversal forming a second triangle.

  1. 2 segments intersect, with segments on left & right ends, creating 2 triangles, left & right ends each have 2 arrows, left end labeled x, right end labeled 8, triangle on left, bottom right side labeled 8, triangle on right, top left labeled 6, bottom left labeled 4.

Parallel lines have equal corresponding angles, so .
By shared angles, .
By , the triangles are similar.




Is there enough information to prove that the triangles are similar?

Use the same method as for part (a).

Labels added: In the triangle with sides x, and 8, angle theta is opposite side 8 and angle alpha is opposite side, x. In the triangle with sides 6, 8, and 4, angle beta is opposite side 8 and angle gamma is opposite side 6. Angles alpha and beta are vertical angles.

Vertical angles are equal, so .
Parallel lines have equal alternate interior angles, so .
Triangles are similar by .