Triangle Proportionality Theorem

Learning Objectives State the Triangle Proportionality Theorem and its converse in their own words. Set up a correct proportion for a triangle with a line parallel to one of its sides. Calculate unknown side lengths in a triangle using the theorem. Apply the Converse of the Triangle Proportionality Theorem to determine if two lines are parallel. Solve algebraic problems involving the Triangle Proportionality Theorem. Differentiate between the proportions used for the theorem and those used for similar triangles. Ever wondered how a cartographer creates an accurate map from aerial photos, or how an artist creates realistic perspective? 🗺️ It all comes down to proportional relationships! This tutorial explores the Triangle Proportionality Theorem, a powerful tool within the s...

TermDefinitionExample RatioA comparison of two quantities by division.The ratio of 3 to 4 can be written as 3:4 or as the fraction 3/4. ProportionAn equation stating that two ratios are equal.The equation 1/2 = 4/8 is a proportion because both ratios are equivalent. Parallel LinesTwo or more lines in a plane that never intersect.In triangle ABC, if a line segment DE is drawn such that DE is parallel to BC (written as DE || BC), then DE and BC will never meet. TransversalA line that intersects two or more other lines at distinct points.In triangle ABC, if a line DE intersects sides AB and AC, then DE acts as a transversal to those two sides. Corresponding AnglesAngles that are in the same relative position at each intersection where a straight line crosses two others. If the two lines are...

Triangle Proportionality Theorem (Side-Splitter Theorem) If a line parallel to one side of a triangle intersects the other two sides, then it divides the two sides proportionally. In ΔABC, if DE || BC, then \frac{AD}{DB} = \frac{AE}{EC}. Use this theorem when you know a line is parallel to a side of a triangle and you need to find a missing length of one of the segments it creates on the other two sides. Converse of the Triangle Proportionality Theorem If a line divides two sides of a triangle proportionally, then it is parallel to the third side. In ΔABC, if \frac{AD}{DB} = \frac{AE}{EC}, then DE || BC. Use this theorem to prove that a line segment within a triangle is parallel to the third side. You must first show that the ratios of the divided segments are equal. Cor...