Triangle Proofs - Interactive Activity

A proof is a logical, step-by-step explanation that shows the truth of a hypothesis (the given information) guarantees the truth of the conclusion (what is to be proved). In a formal two-column proof, every statement is paired with a reason — a definition, property, postulate, or theorem — that justifies it. Triangle proofs combine the congruence shortcuts with supporting reasons: a midpoint produces congruent segments, parallel lines produce congruent alternate interior angles, segments that intersect produce congruent vertical angles, and a shared side is congruent to itself. The proof ends with a congruence statement, and CPCTC can then deliver any final part.

Words and Ideas for Proofs

Vocabulary

Proof: a logical, step-by-step explanation that shows the truth of a hypothesis (the given information) guarantees the truth of the conclusion (what is to be proved).
Two-Column Proof: a formal proof organized in two columns: statements on the left, in logical order, and the reason justifying each statement on the right.
Reflexive Property: a side or angle shared by two triangles is congruent to itself, so it can serve as one of the congruent parts a proof needs.
CPCTC: Corresponding Parts of Congruent Triangles are Congruent — the reason used after a congruence statement to conclude that one specific pair of corresponding parts is congruent.

Main ideas

Start a proof with the given information, work through new information step by step, and end with the statement you were asked to prove — every step carries a reason.
Before writing, mark the diagram first: the given parts, the pairs a midpoint creates, and the angles that must be congruent.
Three reliable sources of observable congruent parts: the definition of midpoint (two congruent segments), vertical angles (congruent, equal in measure), and a shared side or angle (reflexive property).
Parallel lines intersected by a transversal give congruent alternate interior angles (equal in measure) — a common way to earn an angle pair in a proof.
Collect three qualifying parts, then finish with a shortcut — SSS, SAS, ASA, AAS, or HL — and use CPCTC only after the congruence statement, if the conclusion asks for one specific part.
Vertical angles and linear pairs are formed at a single intersection. They are always congruent (equal in measure) or supplementary (their measures add to 180°) on their own, and they are never evidence that two lines are parallel.

Observable Parts

What to do: Press one of the four scenario buttons. Each scenario shows its own diagram, and the observable pair — the congruent pair you can read straight from the diagram — is highlighted. Read the readout to see the pair and the formal reason that justifies it.

What to notice: a proof rarely starts from nothing. A midpoint, a single intersection, a shared side, and a pair of parallel lines each hand you one congruent pair before you do any other work.

Step-by-Step Proof

Given: line UH ∥ line RE, and A is the midpoint of segment UE. Prove: △UAH ≅ △EAR, and then segment UH ≅ segment ER.

The diagram shows the given information: line UH is parallel to line RE (the matching marks on those two lines show that they are parallel), segment UE intersects both parallel lines, and A is the midpoint of segment UE, where segments UE and HR intersect. The tick marks on segment UA and segment EA show that those two segments are congruent (equal in length).

What to do: The first two rows of the proof — the given statements — are already in the table. At each step, the next statement is shown and you choose its reason from four choices. Choose the one correct reason. If your first choice is wrong, you will get a hint and another try, and the missed choice is disabled. After a second miss on a step, the explanation appears, the step is completed for you, and it is marked as shown, not earned. Each completed step adds its row to the two-column proof table, which stays on screen. Press Reset Proof to start the proof over.

Given: line UH ∥ line RE, and A is the midpoint of segment UE. Prove: △UAH ≅ △EAR, and then segment UH ≅ segment ER.

Statements	Reasons

Build the Proof

Given: line PR ∥ line QS, and ∠QPS ≅ ∠RSP. Prove: △PQS ≅ △SRP.

The diagram shows line PR parallel to line QS (the matching marks show that they are parallel), and segment PS intersects both parallel lines. The two triangles, △PQS and △SRP, share the side segment PS.

What to do: This is a fresh attempt at a complete proof. The given row is already in the table. At each step, the next statement is shown and you choose its reason from four choices, the same way as in the activity above. Watch out: some choices are tempting but wrong — a reason that belongs later in a proof cannot be used early, and a pair of angles at two different intersections is not a pair of vertical angles. The correct path is the only path that completes the proof. If your first choice is wrong, you will get a hint and another try; after a second miss on a step, the explanation appears and the step is marked as shown, not earned. Press New Proof to restart cleanly and try again.

Given: line PR ∥ line QS, and ∠QPS ≅ ∠RSP. Prove: △PQS ≅ △SRP.

Statements	Reasons

Your Turn

Ready to practice? Work through these six questions, one at a time. If your first answer is wrong, you will get a hint and another try; the full explanation appears once you answer correctly, or after a second miss. Your score counts first-try answers.

Triangle Proofs — Mr. J Geometry. All activities on this page run entirely in your browser.