Geometry
GradesGrade 8GeometryAA Similarity Criterion

AA Similarity Criterion

📐 Geometry
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AA Similarity Criterion

Two triangles with two pairs of equal angles are always similar!

Two triangles are SIMILAR if they have the same shape but not necessarily the same size — corresponding angles are equal and corresponding sides are proportional. The Angle-Angle (AA) Criterion: if two angles of one triangle equal two angles of another, the triangles must be similar (the third angle is forced to match because all angles sum to 180°).

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AA Criterion: Two pairs of equal angles → triangles are similar. Notation: △ABC ∼ △DEF.

🌳Shadows and AA similarity
A 6 ft person casts a 4 ft shadow.
A nearby tree casts a 20 ft shadow.
Both form right triangles with the ground.

Two matching angles: right angle (90°) + sun angle.
AA criterion satisfied!
Proportion: 6/4 = tree height/20
Tree height = 6 × 20 ÷ 4 = 30 ft

Using similar triangles to find unknown lengths:

1Confirm the triangles are similar using the AA criterion
2Match up corresponding vertices (equal angles correspond)
3Set up a proportion: (side of △1) / (corresponding side of △2) = same ratio
4Cross-multiply and solve for the unknown side
ABDE=BCEF=CAFD=k(scale factor)\frac{AB}{DE} = \frac{BC}{EF} = \frac{CA}{FD} = k \quad (\text{scale factor})
✏️ Try It!

Triangle PQR has angles 40°, 60°, 80°. Triangle STU has angles 40°, 80°, 60°. Are they similar?

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Remember This!

Parallel lines cut by a transversal create equal corresponding angles — a classic setup for AA similarity proofs. Look for parallel sides in overlapping triangles!

Take Quiz 📝 — 25 Questions