7.2.1 Applying Properties of Similar Triangles Key Objectives
• Use properties of similar triangles to find segment lengths. • Apply proportionality and triangle angle bisector theorems. Theorems, Postulates, Corollaries, and Properties • Triangle Proportionality Theorem If a line parallel to a side of a triangle intersects the other two sides, then it divides those sides proportionally. • Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. • Two-Transversal Proportionality Corollary If three or more parallel lines intersect two transversals, then they divide the transversals proportionally. • Triangle Angle Bisector Theorem An angle bisector of a triangle divides the opposite side into two segments whose lengths are proportional to the lengths of the other two sides. Example 1 Finding the Length of a Segment
If a line parallel to a side of a triangle intersects the other two sides, then it divides those two sides proportionally. Look carefully at the proportion of segment lengths. The length of an unknown line segment is determined using the Triangle Proportionality Theorem in this example. It is given that PT || QS . Set up a proportion according to the Triangle Proportionality Theorem and substitute the known segment lengths. Cross multiply and solve for the length of the unknown segment, TS . The solution yields TS = 18/7.
Example 2 Verifying Segments are Parallel
If a line divides two sides of a triangle proportionally, then it is parallel to the third side.
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