Key takeaway: AA similarity says that when a pair of triangles share two matching angle pairs, the triangles are similar, so every pair of corresponding sides has the same ratio.
Grade 10 – GeometryG-SRT.A.3Geometry and Measurement
Definition of AA Similarity
AA similarity is the quickest test for similar triangles. Check just two pairs of angles. If they match, the triangles are similar.
Why two are enough: every triangle’s corners total \(180^{\circ}\), so once two corners match, the third corners must match too. Equal angles on all three corners fix the shape completely. The sizes may differ, but every side of the larger triangle is the same multiple of its partner. The rule says nothing about congruence, because no side is compared.
Formula and notation
When the corners at \(A\) and \(B\) equal the corners at \(D\) and \(E\), the triangles are similar:
\[ \triangle ABC \sim \triangle DEF \quad \Rightarrow \quad \frac{AB}{DE} = \frac{AC}{DF} = \frac{BC}{EF} \]
In a proof, write the reason as “AA” or “AA Similarity Postulate.”
Key facts
- Only two pairs of equal angles are needed, because the third pair follows.
- A shared angle or a pair of vertical angles counts as one equal pair.
- Parallel lines cut by a transversal often supply the equal angles.
- It proves similarity, not congruence, since no side lengths are compared.
AA Similarity example
Two ramp braces are right triangles. Brace ABC is square at corner A, with AB = 9 in. and AC = 12 in. Brace DEF is square at corner D, with DE = 15 in., and the corners at B and E have the same size. Show the braces are similar and find DF.
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The corners at A and D are both square, and the corners at B and E match by the given information. That is two pairs of equal corners, so the triangles are similar by AA.
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Ratio of the sides that match: \(\frac{DE}{AB} = \frac{15}{9} = \frac{5}{3}\).
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Then \(DF = \frac{5}{3} \cdot AC = \frac{5}{3}(12) = 20\).
The triangles are similar by AA, and \(DF = 20\) in.
Common mistake
Watch out
Because the two braces share two pairs of equal corners, Kira concludes they are congruent, the same size and shape.
Why it is wrong
Equal angles prove similarity, not congruence. The sides are 9 and 12 in one triangle and 15 and 20 in the other, so the sizes differ. Congruence needs a side comparison as in SSS, SAS, ASA or AAS.
Related terms and lessons
- Similar Triangles
- Similar Figures
- Triangle Angle Sum Theorem
- Scale Factor
- Triangle Congruence (SSS, SAS, ASA, AAS)
- Dilation
- Geometry · Dilations and Similarity TransformationsLessonWorksheetQuiz
- Geometry · Similar Triangles and ProportionsLessonWorksheetQuiz
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