AA Similarity

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 50 of 60.

Similar triangles have the same shape but not always the same size: matching angles are equal and matching sides are in the same ratio. The sign ∼ means "is similar to". Two angles are enough to prove it, because two angles fix the third and so fix the shape. The rule is AA similarity.

$Triangle ABC and a larger triangle DEF, with arcs at A and D and double arcs at B and E$

Statement	Reason
1. ∠CAB = ∠FDE	Given
2. ∠ABC = ∠DEF	Given
3. △ABC ∼ △DEF	AA similarity, using lines 1 and 2

AA needs no sides at all. That is the difference from ASA: equal angles fix the shape, and only a side pair would fix the size. In a proof the angle pairs often come from earlier rules rather than arcs. On the board, pick the Similar relation instead of Congruent when you tap the two triangles.

Challenge

In △PQR the segment ST is parallel to QR, with S on PQ and T on PR. Prove that △PST ∼ △PQR, then press Check.

Try it yourself

$Write the proof$

Write the proof

All lessons in Congruence & Proof

2Congruent Triangles

Same Size, Same Shape What Congruence Gives Recap: Matching Parts

5ASA and AAS

The Side Between The ASA Rule The AAS Rule ASA or AAS?Recap: ASA and AAS

8Proofs with Vertical Angles

Vertical Angles in Proofs The Bowtie Bowtie with ASA Linear Pairs in Proofs Recap: Vertical Angles

11Similarity Proofs