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This note explores the concepts of similar triangles and the Side-Splitter Theorem, which states that a line parallel to one side of a triangle divides the other two sides proportionally. Additionally, it introduces the Corollary to the Side-Splitter Theorem, applicable when three parallel lines intersect two transversals. The Triangle-Angle Bisector Theorem is also discussed, demonstrating how an angle bisector divides the opposite side into segments proportional to the other two sides. Examples and practical applications for solving for unknown lengths using these principles will be provided.
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Proportions in Triangles Unit 4.11 Notes
Remember the similar triangles that look like this: Notice: Parallel Lines
Remember the similar triangles that look like this: The length of TR is x+16 The length of TV is 15 Previously, we had to find the length of the small triangle’s sides and the big triangle’s sides to solve for x. But we can actually get around that step. What do you think our proportion might look like? x 5 = 16 10
Side-Splitter Theorem If a line is parallel to one side of a triangle and intersects the other two sides, then it divides those sides proportionally. d b = a c =
Do you think the whole shape actually has to be a triangle for this theorem to work? = Nope! It will still work as long as the lines are parallel!
Corollary to the Side-Splitter If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional. =
Example Solve for x and y by using proportions with the lengths of the transversal segments in the middle.
Triangle-Angle-Bisector Theorem If a ray bisects an angle of a triangle, then it divides the opposite side into two segments that are proportional to the other two sides of the triangle. =
Solve for x. Example
2) 1) Solve for x in the three examples above. 3) Exit Ticket
