A proofis a logical argument that starts from what’s “given” (known facts about the figure) and ends at what you’re asked to show, with every step justified by a definition, postulate, or previously proven theorem. The most common format in an introductory course is the two-column proof: statements on the left, reasons on the right.
Let me show you one, start to finish
Given: Lines AB and CD intersect at point E. Prove:∠AEC ≅ ∠BED (the vertical angles are congruent).
| Statement | Reason |
|---|---|
| 1. Lines AB and CD intersect at E. | 1. Given |
| 2. ∠AEC and ∠AED are supplementary. | 2. Angles on a straight line (CD) sum to 180° |
| 3. ∠AED and ∠BED are supplementary. | 3. Angles on a straight line (AB) sum to 180° |
| 4. ∠AEC ≅ ∠BED | 4. Both are supplementary to ∠AED, so they’re equal |
Notice every single line follows directly from the one before it. No step requires seeing the whole proof in advance — each one only asks, “given what I just established, what can I now conclude?”
Where to start: write down what’s given and what you want
Before attempting any steps, write the “given” information and the statement to prove at the top, in your own words. Then mark the diagram with everything given — tick marks for equal sides, arcs for equal angles. Most stuck proofs are stuck because a given fact never got translated onto the figure, not because the logic itself is too hard.
The mistake I see most
Trying to jump straight from the given information to the final statement, skipping the small intermediate facts that connect them. The fix is asking, after every single step, “what does this one new fact let me say next?” — and writing that down as the very next line, even if it doesn’t look like it’s getting you anywhere yet. It usually is.
Want to work through it together?
If proofs still feel like trying to see the ending before you’ve taken the first step, that’s the exact feeling we work on directly — building the habit of trusting one small, justified step at a time.
