Free tool
Geometry Proof Practice
A two-column proof pairs each statement with the reason that justifies it, working from the given information to what you must prove. The statements below are already in order - you supply the reasons, which is the column that actually earns marks. Six classic proofs, instant feedback, free.
Vertical angles are congruent
- Given:
- Lines AB and CD intersect at point E
- Prove:
- ∠AEC ≅ ∠BED
The statements are already in order. Your job is the part that actually gets marked: choose the reason that justifies each line.
| # | Statement | Reason |
|---|---|---|
| 1. | Lines AB and CD intersect at point E | |
| 2. | ∠AEC and ∠AED form a linear pair; ∠AED and ∠BED form a linear pair | |
| 3. | m∠AEC + m∠AED = 180° and m∠AED + m∠BED = 180° | |
| 4. | m∠AEC + m∠AED = m∠AED + m∠BED | |
| 5. | m∠AEC = m∠BED | |
| 6. | ∠AEC ≅ ∠BED |
How a two-column proof works
Every proof is a chain. You begin with the given facts, and each new line must follow from something already on the page - a definition, a postulate, an algebraic property, or a theorem you have already proven. You finish when the last statement is exactly the thing you were asked to prove.
The reliable way to build one is to work from both ends. Write the given information at the top and the goal at the bottom, then ask what single fact would make the goal true. Very often that fact is "these two triangles are congruent", which turns the whole problem into: which three parts can I match? Once the triangles are congruent, CPCTC finishes the job in one line.
Two moves account for a surprising share of lost marks. The first is the shared side: when two triangles overlap or share a diagonal, that side is congruent to itself by the Reflexive Property - it is a free matching part students routinely forget to claim. The second is switching between congruence and equality: ≅ is for figures, = is for numbers, and you move between them with the definition of congruent segments or angles.
The reasons that justify most proof lines
Nearly every line in a high-school geometry proof is justified by one of these:
| Reason | Use it when |
|---|---|
| Given | The fact was handed to you in the problem statement. |
| Definition of midpoint | A midpoint splits a segment into two congruent segments. |
| Definition of congruent segments | Move between AB ≅ CD (geometry) and AB = CD (arithmetic). |
| Definition of congruent angles | Move between ∠A ≅ ∠B and m∠A = m∠B. |
| Reflexive Property of Congruence | A shared side or angle is congruent to itself (AC ≅ AC). |
| Vertical Angles Theorem | Two angles opposite each other where two lines cross are congruent. |
| Linear Pair Postulate | Two angles forming a straight line are supplementary (sum 180°). |
| Corresponding Angles Postulate | Parallel lines cut by a transversal make congruent corresponding angles. |
| Alternate Interior Angles Theorem | Parallel lines cut by a transversal make congruent alternate interior angles. |
| Segment Addition Postulate | If B lies between A and C, then AB + BC = AC. |
| SSS / SAS / ASA / AAS Congruence | Prove two triangles congruent from three matching parts. |
| CPCTC | After proving triangles congruent, conclude any matching part is congruent. |
Frequently asked questions
What is a two-column proof?+
A two-column proof is a geometry argument written as a table: each row has a statement on the left and the reason that justifies it on the right. You start from the given information and end at what you were asked to prove, and every single line must be justified by a definition, postulate, property, or previously proven theorem.
Why are geometry proofs so hard?+
Because the statements usually feel obvious while the reasons do not. Most students can see that two angles are equal but cannot name why, and the reason column is what actually earns marks. That is why this tool gives you the statements in order and asks only for the reasons - it drills the exact step people lose points on.
What reason do I use when a side is shared by two triangles?+
The Reflexive Property of Congruence. A segment or angle is always congruent to itself, so a side shared by two triangles - like the diagonal AC in a parallelogram - is congruent to itself and counts as one of your matching parts.
What does CPCTC mean and when can I use it?+
CPCTC stands for Corresponding Parts of Congruent Triangles are Congruent. You may only use it after you have already proven the two triangles congruent by SSS, SAS, ASA, or AAS. It is almost always the last line of a proof, used to conclude that a specific pair of sides or angles matches.
Is this geometry proof practice free?+
Yes - it runs entirely in your browser, needs no signup, and stores nothing. Work through all six proofs as many times as you like.
Still stuck on proofs?
Proofs are the point where geometry stops being about shapes and starts being about argument - and it is much faster to learn that with someone watching your reasoning. Your first trial lesson is free.
Turn proofs into the easiest marks on the paper
Work through proofs live with a geometry specialist who shows you how to find the next line, not just the answer - your first trial lesson is free.