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Math · Geometry

Triangle postulates and theorems

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.

Practise triangle postulates and theorems nowSee sample questions
3
Difficulty tiers
40s
Target pace
51%
Average first-attempt score

What's covered

  • SSS, SAS, ASA congruence — identify which combination of given sides and angles proves two triangles congruent, and recognize when a combination — like SSA — proves nothing.
  • Similarity postulates — apply AA, SAS, and SSS similarity to establish that two triangles have proportional sides, even without matching size.
  • The triangle inequality — any two sides of a triangle must sum to more than the third side — use this to check whether three lengths can even form a triangle.
  • Angle sum of a triangle — the three interior angles of any triangle always sum to exactly 180°, regardless of the triangle's shape.
  • Corresponding parts of congruent triangles — once two triangles are proven congruent, every corresponding side and angle between them is automatically equal too.

Where students lose marks

Treating SSA as a valid congruence shortcut

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.

Matching sides or angles out of order

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.

Forgetting to check the triangle inequality before assuming three lengths form a triangle

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.

Three sample questions

Straight from the bank — one per difficulty tier. Reveal the answer to see the explanation you'd get in a real session.

Sample 01Foundation

Two triangles share two pairs of congruent corresponding sides, and the included angle between them is also congruent. Which postulate proves the triangles congruent?

Sample 02Core

Can a triangle have side lengths of 4 cm, 9 cm, and 5 cm?

Sample 03Advanced

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.

How to practise this

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.

Name the postulate before solving

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.

Then 40 seconds a question

Move to timed sessions once postulate identification is instant. Triangle problems run longer since there's often an equation to set up first.

Fold into a mock

Pair this with basic definitions and circles in a mock — all three build on the same angle-relationship foundation.

Stop guessing the postulate.
Start naming it first.

  • A postulate-by-postulate walkthrough on every item
  • Timed or untimed sessions, any length you like
  • Tracked separately — see your triangle postulates accuracy and pace over time
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Related Math subtopics

Area and perimeter of basic shapesBasic definitions: angles, lines, linear pairsCirclesSupplementary, complementary, congruentAll of Math