SAS Similarity

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 53 of 60.

The last similarity rule mirrors SAS. If two sides of one triangle are in the same ratio as two sides of another, and the angles between them are equal, the triangles are similar: SAS similarity. Its line cites the equal angles; the side lengths come from the figure.

$Triangles ABC and DEF: AB = 5, CA = 4, DE = 15, FD = 12, with the angles at A and D marked equal$

Statement	Reason
1. ∠BAC = ∠EDF	Given
2. △ABC ∼ △DEF	SAS similarity, using line 1

Check both parts: 15/5 = 3 and 12/4 = 3, so the ratios match, and the marked angles at A and D sit between those sides. With the angle anywhere else, the rule would not apply, just as SSA proves nothing for congruence.

Challenge

△GHK has GH = 4 and KG = 3; △LMN has LM = 8 and NL = 6. The angles at G and L are marked equal. Prove that △GHK ∼ △LMN, 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