Grades Geometry & Functions Similarity & TrigonometrySimilar Triangles & the AA Criterion

Similar Triangles & the AA Criterion

📐 Similarity & Trigonometry

🔍

Similar Triangles & the AA Criterion

Prove triangles similar and use proportional sides to solve problems.

Two triangles are similar if their corresponding angles are equal and their corresponding sides are proportional. The AA (Angle-Angle) criterion states that if two angles of one triangle equal two angles of another, the triangles are similar.

\triangle ABC \sim \triangle DEF \implies \frac{AB}{DE} = \frac{BC}{EF} = \frac{AC}{DF} = k \quad (\text{scale factor } k)

Using Similarity to Find Missing Sides

1Identify the two similar triangles and their correspondence.

2Write the proportion of corresponding sides.

3Cross-multiply and solve for the unknown.

4Verify the answer is reasonable given the scale factor.

📏Solving with Similar Triangles

Triangles ABC ~ DEF with AB = 6, BC = 9, DE = 4. Find EF. Set up proportion: 6/4 = 9/EF → EF = 9·4/6 = 6.

\frac{AB}{DE} = \frac{BC}{EF} \implies \frac{6}{4} = \frac{9}{EF} \implies EF = 6

💡

Remember This!

AA is the most efficient similarity criterion because you only need two pairs of angles — the third pair is automatically equal since all angles in a triangle sum to 180°.

✏️ Try It!

If △ABC ~ △DEF with a scale factor of 3:2, and AB = 15, what is the length of DE?

Take Quiz 📝 — 25 Questions