Similarity & Ratios

Angles give you lengths.

Similarity is the bridge from angle information to length information, and AA is all you need to cross it. A parallel line is the most common trigger: $DE \parallel BC$ instantly gives $\triangle ADE \sim \triangle ABC$ .

Keep the companions close: the angle bisector theorem ( $BD/DC = AB/AC$ ), the midsegment (parallel and half), and the altitude to a hypotenuse, which produces the geometric mean $AH^2 = BH \cdot HC$ .

One trap causes more lost marks than any other here: areas scale as the square of the length ratio, never the ratio itself.

Where it appears on Lemma: Level 3 (similar triangles), and inside most length-chasing problems.

Train it on Lemma

Similarity & Ratios unlocks at Level 3 of Lemma's eight-level ladder, with lessons that teach it and drills that make it stick. 78 lessons, 624 curated problems and unlimited generated practice at six difficulties. Free to start — no card, no trial clock.

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