The geometry solver that actually teaches you.

Why geometry needs a different kind of solver

Short answer: capture the diagram and the givens together, in one screenshot. Geometry is the one topic where half the problem is drawn rather than written — the tick marks, the right-angle squares, the shared side two triangles have in common. Crop to the text alone and the problem becomes unsolvable, because the information that licenses each step was in the picture.

Algebra solvers take an expression and transform it. Geometry does not work that way. A geometry problem is a set of given facts plus a figure, and the answer is a chain of justified deductions. That is why a calculator that returns "x = 8" is close to useless in a geometry class: the number was never what you were being graded on.

Two-column proofs

A proof is a sequence of statements where every line carries the reason it is allowed. Most students can see that two triangles are congruent; the marks are lost in naming why. The reasons available to you are a short and finite list, and knowing which one applies is the whole skill:

• Given — stated in the problem. Every proof starts here.
• Reflexive property — a segment or angle is congruent to itself. This is how a shared side enters the proof, and it is the single most commonly forgotten line.
• Vertical angles are congruent — free whenever two lines cross.
• Alternate interior / corresponding angles — available only once you have established parallel lines.
• Definition of midpoint, bisector, perpendicular — converts a named object into the equality you can actually use.
• CPCTC — corresponding parts of congruent triangles are congruent. It comes after a congruence conclusion, never before it. Using it to prove the congruence itself is circular, and graders look for exactly that.

Which congruence justification applies: SSS, SAS, ASA, AAS

Four work, and two look like they should but do not.

• SSS — all three pairs of sides congruent.
• SAS — two sides and the angle between them. The angle must be included; two sides and some other angle proves nothing.
• ASA — two angles and the side between them.
• AAS — two angles and a side that is not between them. Valid, because two angles fix the third.
• AAA — not congruence. It proves similarity only: same shape, any size.
• SSA — not a valid justification. Two sides and a non-included angle can describe two different triangles. This is the ambiguous case, and it is the trap the exam is testing.

Work out which one you have by marking the diagram before writing anything: tick the sides you know are equal, arc the angles, and mark the shared side. The pattern that emerges names the justification for you. For right triangles, HL (hypotenuse–leg) is also available.

Paragraph proofs

A paragraph proof contains exactly the same logic as a two-column proof, written in prose. Nothing may be dropped — each sentence still carries its justification, usually as a "since" or "because" clause. The usual way to lose marks is to write the narrative and quietly leave out the reasons, because prose makes the gap easier to hide. Draft it in two columns first, then translate line by line.

Not losing marks on proofs

• Never justify a step from how the diagram looks. Two segments that appear equal are not equal until something says so. Figures are not drawn to scale unless stated.
• Claim the reflexive side. When two triangles share an edge, that congruence needs its own line.
• Get CPCTC's order right. Prove congruence first, then use CPCTC to conclude the parts.
• Match the correspondence. △ABC ≅ △DEF means A↔D, B↔E, C↔F. Writing the vertices in the wrong order makes every part-wise conclusion wrong.
• State the parallel before you use the angle rule. Alternate interior angles are only congruent once the lines are known to be parallel.
• End on the thing you were asked to prove, stated in full.

More on choosing a strategy — working backwards from the goal, and which theorem to reach for — in geometry proof strategies for students.

Beyond proofs

Triangles: angle sums, the triangle inequality, similarity ratios, and the Pythagorean theorem — there is a dedicated Pythagorean theorem calculator with exact radical answers, plus worked practice problems. Circles: central and inscribed angles, tangent and chord relationships, arc length and sector area. Coordinate geometry: distance, midpoint, slope, and proving a quadrilateral is what the question claims. Solids: surface area and volume, including the composite figures that are mostly bookkeeping.

The geometry guide covers the theorem list in full.

Capturing a geometry problem properly

1. Include the figure. The diagram carries the tick marks, right-angle squares and labels that the text assumes you can see.
2. Include the givens. "Given: AB ≅ CD" is load-bearing — without it the problem genuinely has no solution.
3. Include the instruction line. "Prove", "find x", and "explain why" want different answers.
4. Keep the whole figure in frame. A cropped vertex label turns a solvable problem into a guess.

Working from a textbook or a handwritten worksheet rather than a screen? See solving handwritten math from a photo. Already finished a proof and want it checked rather than solved — which is the better way to use this on graded work — that is the math checker.

Free geometry help, then what

The first 20 problems are free, with full reasoning, no credit card and no account needed to start. Teach Me mode asks you which justification applies instead of supplying it, which is the version worth using the night before a proofs test. After that, Lite ($1.99/mo) covers a weekly problem set and Student ($4.99/mo) covers a semester — see pricing.