Similar Triangles and Proportional Sides

Objective: Use one pair of corresponding sides to find another.

Prerequisite skills: proportions and corresponding triangle parts. Similar triangles have equal corresponding angles and proportional corresponding side lengths. A scale factor multiplies every side from the first triangle to the second; it is not added to each length. Common error: adding the change in one side to every other side. Restrictions and interpretation: AA, proportional SSS, or proportional SAS with included angle can establish similarity; corresponding side ratios must use one consistent direction.

Objective: Use two matching angles to establish similarity without size equality.

Two corresponding equal angles force the third equal by the triangle sum, establishing AA similarity. SSS similarity requires proportional triples, not necessarily equal side lengths. Shape is fixed, while scale can vary. Common error: requiring equal side lengths to prove similarity. Restrictions and interpretation: AA, proportional SSS, or proportional SAS with included angle can establish similarity; corresponding side ratios must use one consistent direction.

Objective: Use parallelism to establish similarity before a proportion.

A segment parallel to a triangle side creates a smaller triangle sharing one angle and corresponding parallel-line angles. Use the matching side ratios, then distinguish a complete side length from a remaining segment. Common error: using DB as though it were the full AB length. Restrictions and interpretation: AA, proportional SSS, or proportional SAS with included angle can establish similarity; corresponding side ratios must use one consistent direction. Desmos and method choice: Use Desmos for arithmetic or a graph/table check when that reduces work. Enter parentheses explicitly and verify the requested quantity and units. Use the algebraic steps shown here for an exact result; a rounded graph display is not proof of exact equality.