Similarity rules for triangles

Learning Objectives Identify the conditions required for the Angle-Angle (AA), Side-Angle-Side (SAS), and Side-Side-Side (SSS) similarity rules. Apply the AA, SAS, and SSS similarity rules to determine if two triangles are similar. Write a correct similarity statement (e.g., ΔABC ~ ΔXYZ) that reflects the correspondence between vertices. Set up and solve proportions to find unknown side lengths in similar triangles. Use the properties of similar triangles to find unknown angle measures. Construct a formal proof to demonstrate that two triangles are similar. How can you measure the height of a giant redwood tree using only its shadow and a meter stick? 📏 The secret lies in the power of similar triangles! This tutorial will introduce the three fundamental shortcuts for provi...

TermDefinitionExample Similar TrianglesTriangles that have the same shape but not necessarily the same size. This means their corresponding angles are congruent (equal) and their corresponding sides are in proportion.If ΔABC ~ ΔXYZ, then ∠A ≅ ∠X, ∠B ≅ ∠Y, ∠C ≅ ∠Z, and AB/XY = BC/YZ = AC/XZ. Corresponding PartsThe angles and sides of two triangles that are in the same relative position. In a similarity statement like ΔABC ~ ΔXYZ, ∠A corresponds to ∠X, and side AB corresponds to side XY.In ΔPQR ~ ΔLMN, the side corresponding to PR is LN. Similarity Ratio (or Scale Factor)The constant ratio between the lengths of corresponding sides of two similar triangles.If ΔABC has sides 3, 4, 5 and similar ΔXYZ has sides 6, 8, 10, the similarity ratio of ΔABC to ΔXYZ is 3/6 = 1/2. ProportionAn equation...

Angle-Angle (AA) Similarity Postulate If ∠A ≅ ∠X and ∠B ≅ ∠Y, then ΔABC ~ ΔXYZ. If two angles of one triangle are congruent to two angles of another triangle, the triangles are similar. This is the most frequently used similarity rule. Side-Angle-Side (SAS) Similarity Theorem If ∠A ≅ ∠X and (AB/XY) = (AC/XZ), then ΔABC ~ ΔXYZ. If an angle of one triangle is congruent to an angle of another triangle and the lengths of the sides that include these angles are proportional, the triangles are similar. Side-Side-Side (SSS) Similarity Theorem If (AB/XY) = (BC/YZ) = (AC/XZ), then ΔABC ~ ΔXYZ. If the corresponding side lengths of two triangles are all proportional, the triangles are similar.