SSS Similarity

Part of the Congruence & Proof section of Coddy's Geometry journey. Lesson 55 of 63.

Similar triangles can be proved from their sides too. If all three sides of one triangle are the same multiple of the sides of another, the triangles are similar. This is SSS similarity. Its line reads the lengths from the figure, so it cites nothing; the work is finding the matching.

$Triangle PQR with sides 3, 5, 6 and triangle XYZ with sides 9, 18, 15$

Sort each triangle's sides by length and pair them up: 3 with 9, 5 with 15 and 6 with 18. Every pair has the ratio 3, so the shapes match. Now read the corners: PQ (3) pairs with XY (9) and QR (5) with ZX (15), so Q, the corner the two short sides share, matches X. In the same way P matches Y and R matches Z: △PQR ∼ △YXZ.

Challenge

△ABC has sides AB = 4, BC = 6 and CA = 8; △DEF has DE = 12, EF = 16 and FD = 8. The triangles are similar. Write the similarity with the corners in matching order, give its reason, then press Check.

Try it yourself

$Write the proof$

Write the proof

Test yourself

This lesson includes a short quiz. Start the lesson to answer it and track your progress.

All lessons in Congruence & Proof

2Congruent Triangles

Same Size, Same Shape Matching Sides Matching Angles 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