Start free
  • Euclidean Geometry: Triangles and Circles

How do you prove two triangles are congruent?

Quick answer

To prove two triangles are congruent, show that three specific corresponding pairs of sides and angles are equal using an established congruence theorem: Side-Angle-Side (SAS), Angle-Side-Angle (ASA), Side-Side-Side (SSS), or Side-Side-Angle where the angle faces the longer side (such as Hypotenuse-Leg). Once proved, all remaining corresponding parts are equal (CPCTC).

The topic explained

The four main congruence criteria

Two triangles are congruent when they have the exact same size and shape. You do not need to measure all six parts (three sides and three angles) to prove this. Instead, you only need to match three specific pairs using a geometric theorem.

The standard criteria are Side-Angle-Side (SAS), where the equal angle sits strictly between the two equal sides; Angle-Side-Angle (ASA), where the side lies between two known angles; Side-Side-Side (SSS), where all three pairs of corresponding sides match; and Side-Side with the angle opposite the longer side (SsA). In right triangles, this last case is known as Hypotenuse-Leg (HL).

Writing a two-column proof step by step

A formal geometric proof uses two columns: Statements on the left and Reasons on the right. Start by listing your given information. Next, find unstated shared features, such as a shared side (Reflexive Property) or vertical angles created by intersecting lines.

Once you collect three matching elements that satisfy a criterion like SAS or SSS, state that the two triangles are congruent. Write your congruence statement with vertices in matching order, such as Triangle ABC is congruent to Triangle DEF.

Using CPCTC to finish the proof

Many geometry problems do not stop at proving the triangles congruent. They ask you to prove that a specific segment or angle is equal, such as segment AB = segment DE.

After establishing triangle congruence, use the reason 'Corresponding Parts of Congruent Triangles are Congruent' (CPCTC). CPCTC lets you conclude that any remaining unmeasured matching sides or angles must also be equal.

Free printable worksheet

Triangle Congruence and Proofs Practice

For 10th-grade geometry students. Identify the congruence theorem for each problem, or write the missing statement and reason.

Print the worksheet Regenerate new worksheet

Free, no sign-up. The answer key prints on the second page. Regenerating a new worksheet on the same topic needs a free account.

Free printable worksheetlessoons.com
Triangle Congruence and Proofs PracticeName:

  1. Triangle ABC and Triangle DEF have AB = DE, BC = EF, and AC = DF. State the theorem that proves them congruent.

  2. In Triangle JKL and Triangle MNO, JK = MN, angle K = angle N, and KL = NO. State the theorem that proves them congruent.

  3. Triangle PQR and Triangle STU share angle P = angle S, angle Q = angle T, and segment PQ = segment ST. State the congruence theorem.

  4. Two right triangles share congruent hypotenuses and one pair of congruent legs. Which specific congruence criterion proves them congruent?

  5. Segments WY and XZ bisect each other at point M. To prove Triangle WMX is congruent to Triangle YMZ, you identify WM = YM and XM = ZM. What is the third pair of parts and the reason?

  6. You have proved Triangle ABC is congruent to Triangle XYZ. What is the formal reason to conclude that angle C is congruent to angle Z?

More topics and worksheets at lessoons.com© 2026 Lessons

Common questions

Why is Angle-Angle-Angle (AAA) not a congruence theorem?

Equal angles preserve shape but not size. For example, an equilateral triangle with side length 2 has the same 60-degree angles as an equilateral triangle with side length 10. They are similar, not congruent.

Why does Side-Side-Angle (SSA) usually fail?

If the angle is not between the two known sides, you can often swing the second side into two different positions, creating two non-congruent triangles. SSA only works when the angle lies opposite the strictly longer side, as in right triangles.

What is the most common mistake when using SAS?

The most frequent error is picking an angle that is not included between the two known sides. For SAS, the angle must be formed directly by the two segments you measured.

What should I look for if a diagram does not give three pairs of markings?

Look for shared sides (Reflexive Property), vertical angles where two straight lines intersect, or alternate interior angles created by given parallel lines.

Still have a question? Ask Lessi

Any question about this subject. Lessi answers right here, step by step.

Try asking

Asking needs a free account; your question is kept while you sign up. Answers are written by AI and can contain mistakes. Lessi doesn’t give medical or legal advice.

Learn it with a teacher

Learn this topic live with Lessi

Lessi, the AI teacher, explains it out loud on a board, with short quizzes, at your pace. The first 60 minutes are free.

Prefer a private teacher?

Private teachers for Mathematics

Teachers on Lessoons, online or face to face. Each profile opens on the Lessoons site.

See all 5,572 teachers