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Objectives: Use Side-splitter Theorem and the Triangle-Angle-Bisector Theorem

This section explores the application of ratios and proportions in solving real-world problems involving triangles and polygons. Key concepts include the Side-Splitter Theorem, which states that a line parallel to one side of a triangle divides the other sides proportionally, and the Triangle-Angle Bisector Theorem, which asserts that an angle bisector divides the opposite side into segments proportional to the adjacent sides. The section provides examples related to sailmaking and offers problems requiring the application of these theorems to find unknown lengths.

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Objectives: Use Side-splitter Theorem and the Triangle-Angle-Bisector Theorem

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  1. Section 8-5 Proportions in Triangles SPI 22B: apply ratio and proportion to solve real-world problems involving polygons • Objectives: • Use Side-splitter Theorem and the Triangle-Angle-Bisector Theorem Similar triangles can be used to solve a variety of problems.

  2. Side-Splitter Theorem Theorem 8-4 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.

  3. Apply the Side-Splitter Theorem Find the length of VX by using the side splitter theorem.

  4. Corollary to Side-Splitter Theorem Theorem 8-4 Corollary to Side-Splitter Theorem If three parallel lines intersect two transversals, then the segments intercepted on the transversals are proportional.

  5. Apply the Corollary to the Side-Splitter Theorem Find the value of x from the diagram below.

  6. Real-world Connection Sail makers sometimes use a computer to create patterns for sails. After the panels are cut out, they are sown together to form the sail. The edges of the panels in the sail to the right are parallel. Find the lengths of x and y.

  7. Triangle-Angle Bisector Theorem Theorem 8-5 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. CA • DB = CD • BA

  8. Using the Triangle-Angle Bisector Theorem Find the unknown value for the given information.

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