Apply congruence and similarity postulates — SSS, SAS, ASA — and the triangle inequality to prove figures congruent and solve for missing sides or angles. Every question comes with a written explanation of exactly which postulate justified the answer.
Two sides and a non-included angle don't guarantee congruent triangles — SSA can produce two different triangles from the same given measurements, unlike SSS, SAS, or ASA.
Congruence and similarity statements list corresponding vertices in order — triangle ABC ≅ triangle DEF means A matches D, B matches E, C matches F, not just any pairing that happens to fit.
Three lengths where the two shortest don't sum to more than the longest can't form a triangle at all, no matter how the problem describes them.
Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.
Two triangles share two pairs of congruent corresponding sides, and the included angle between them is also congruent. Which postulate proves the triangles congruent?
A — SAS. Side-Angle-Side (SAS) congruence requires two sides and the angle between them to be congruent, exactly what's described. AA (D) only proves similarity, not congruence — it doesn't guarantee the triangles are the same size.
Can a triangle have side lengths of 4 cm, 9 cm, and 5 cm?
A — No — 4 + 5 is not greater than 9. The triangle inequality requires the sum of any two sides to be strictly greater than the third: 4 + 5 = 9, which is not greater than the third side, 9 — it's exactly equal. These three lengths can't form an actual triangle; they'd collapse into a straight line. Yes — 4 + 9 is greater than 5 (D) checks only one of the three required pairwise comparisons and stops, missing the one combination that actually fails.
In triangle ABC, angle A measures 2x, angle B measures 3x + 10, and angle C measures 5x − 40. Find the measure of the largest angle.
A — 73°. The three angles of a triangle sum to 180°: 2x + (3x+10) + (5x−40) = 180, giving 10x − 30 = 180, so x = 21. Substituting back: angle A = 42°, angle B = 73°, angle C = 65° — the largest is angle B at 73°. 84° (D) comes from doubling angle A (2x = 42°, doubled to 84°) instead of correctly identifying which of the three computed angles is actually the largest.
Triangle proofs and calculations both come down to matching the right postulate to what's actually given — the arithmetic afterward is usually the easy part.
For every problem, state which congruence or similarity postulate applies — or whether the triangle inequality is in play — before doing any calculation. Untimed practice is where that identification gets fast.
Move to timed sessions once postulate identification is instant. Triangle problems run longer since there's often an equation to set up first.
Pair this with basic definitions and circles in a mock — all three build on the same angle-relationship foundation.
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