Vertical Angles are Congruent

A proof is an argument where nothing is taken on faith: every statement comes with a reason that justifies it, and every step follows from the ones already written above it. The two-column format keeps you honest — statements on the left, the reason for each on the right.

We'll build the classic first proof: when two lines cross, the two angles opposite each other — the vertical angles — are always congruent. You can see it is true; the skill is writing down why, step by step, so the conclusion is forced rather than merely observed.

The idea

Two crossing lines make four angles. Each angle and the one right next to it form a linear pair along a straight line, so together they measure $180°$ . Two of those $180°$ equations share the same in-between angle — subtract it from both and the two vertical angles are left equal. That is the whole argument; the builder below lets you assemble it.

Build the proof

Read the Given and Prove at the top, then fill in the reason that justifies each statement. Click an angle in the figure to find where it appears in the proof, or click a statement to see which angles it is about. Only a step that genuinely follows will lock into place.

Given: Two lines intersect, forming $\angle 1$ , $\angle 2$ , and $\angle 3$

Prove: $\angle 1 \cong \angle 3$

Statements	Reasons
1.
2.
3.
4.
5.
6.

Row 1: choose the reason that justifies this statement.

Reasons

Placed 0 of 6 steps.

Two lines cross, forming the numbered angles shown. Build a two-column proof that

\angle 1 \cong \angle 3

: each statement is given in order and you supply the reason that justifies it. Each row is checked against the logic of the proof, so only a step that genuinely follows will seat.

Key result

A two-column proof pairs every statement with a reason. To prove vertical angles congruent, use the Linear Pair Postulate twice (each pair sums to $180°$ ), substitute to set the two sums equal, then subtract the shared angle — leaving the vertical angles equal. The figure only suggests the result; the reasons are what prove it.

For teachers

Standard, common misconceptions, and suggested use will live here.