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🏰 Geometry Kingdom · Geometry

Triangle Congruence

Decide when two triangles are congruent from the SSS, SAS, ASA, AAS and HL criteria, match corresponding parts, and supply the missing statement or reason in a congruence proof.

In short

  • A congruence statement is read down the columns: "triangle ABC is congruent to triangle DEF" pairs A with D, B with E and C with F, so AB matches DE and angle B matches angle E — even when the second name is written out of alphabetical order.
  • Exactly five patterns prove congruence: SSS, SAS, ASA, AAS and HL. In SAS the angle is between the two sides; in ASA the side is between the two angles; in AAS it is outside them.
  • SSA and AAA are not criteria. SSA leaves two possible triangles, and AAA fixes the shape but not the size, so it proves similarity only.
  • CPCTC comes after the congruence is proved, never before: once the triangles are congruent, all six pairs of corresponding parts are equal, which is what lets you transfer a length or an angle.
  • A side shared by both triangles equals itself (the Reflexive Property) and counts as a full third pair — it is the most commonly missed given in the whole topic.

Congruent means one lays exactly on the other

Two triangles are congruent when one can be picked up, turned, flipped and set down exactly on top of the other. Same shape, same size, no stretching. That is stronger than *similar*, which allows one to be an enlargement of the other.

Because they land on top of each other, every part matches a part: three pairs of sides and three pairs of angles, six matches in all. The whole subject lives on writing those matches down correctly, and there is a rule for that.

The letters of a congruence statement are matched in the order they are written. "Triangle ABC is congruent to triangle DEF" is not a vague claim that the two triangles are the same. It is six statements at once, read down the columns:

A B C D E F

  • A matches D, B matches E, C matches F
  • AB matches DE, BC matches EF, AC matches DF
  • angle A = angle D, angle B = angle E, angle C = angle F

Now watch what happens when the letters are shuffled. "Triangle ABC is congruent to triangle FDE" pairs A with F, B with D and C with E, so AB matches FD (the same segment as DF) and angle B matches angle D. The triangles are the same triangles; only the *bookkeeping* changed. Questions here often scramble the second name on purpose, so always write the two names one above the other before you answer.

At the Mirror Library, Archivist Lumen keeps two copies of every map for exactly this reason: a copy is only useful if you know which corner of it answers to which.

The five criteria — and the two impostors

Checking all six pairs would be slow, and it turns out you never have to. Three of the right kind are enough. There are exactly five patterns that work:

  • SSS — three pairs of matching sides.
  • SAS — two pairs of matching sides and the pair of angles between them.
  • ASA — two pairs of matching angles and the pair of sides between them.
  • AAS — two pairs of matching angles and a pair of sides outside them.
  • HL — in right triangles only: the right angles, the hypotenuses, and one pair of legs.

The word "between" is doing all the work, and it is where most marks are lost. In SAS the angle is the corner where the two given sides meet — that is called the included angle. In ASA the side joins the two given angles. Read the pattern, then check the arrangement.

Two famous patterns are not criteria:

  • SSA — two sides and an angle that is *not* between them. Swing the third side around and there are two different triangles that fit the same three measurements. Not enough.
  • AAA — three pairs of matching angles. Equal angles fix the *shape* but say nothing about the *size*, so AAA proves the triangles are similar, never congruent.

If a set of givens matches SSA or AAA, the honest answer is that there is not enough information. Being able to say so is part of the skill, and it is the right answer just as often as a criterion is.

CPCTC: what a proof buys you

The criteria are a bargain. You hand over three matching pairs and you receive all six, because congruent triangles match in every part. That is abbreviated CPCTC: corresponding parts of congruent triangles are congruent.

The order matters. CPCTC is always used *after* the triangles have been proved congruent, never as a reason for one of the three pairs you used to prove it. In a two-column proof it is the line below the conclusion, not above it.

CPCTC is what turns congruence into arithmetic. If triangle ABC is congruent to triangle DEF and AB = 7 cm, then DE = 7 cm — you did not measure DE, you deduced it. And if two corresponding parts are given as expressions, they must be equal, so you get an equation:

AB = 2x + 3 and DE = 4x - 5 2x + 3 = 4x - 5 8 = 2x x = 4

Read the question before you stop. If it asks for x, the answer is 4. If it asks for AB, put x back in: 2(4) + 3 = 11 cm. Stopping at x when the length was wanted is the single most common slip in this topic, and it is worth one deliberate glance at the question every time.

Reading a two-column proof

A two-column proof is a list of statements, each with a reason. The statements walk from the givens to the conclusion; the reasons say why each step is allowed. A congruence proof almost always uses one of a small set of reasons:

  • Given — it was handed to you.
  • Reflexive Property — a segment shared by both triangles is equal to itself. Free, and easy to forget.
  • Definition of midpoint — a midpoint cuts a segment into two equal halves.
  • Definition of an angle bisector — a bisector cuts an angle into two equal angles.
  • Vertical Angles Theorem — two crossing segments make equal opposite angles.
  • Alternate Interior Angles Theorem — a segment crossing two parallel segments makes equal alternate interior angles.
  • Definition of perpendicular lines, then all right angles are congruent.
  • One of the five criteria, on the line that concludes the congruence.
  • CPCTC, on any line after that.

Here is a whole proof. In the figure, segments AC and BD cross at M, and M is the midpoint of both.

1. M is the midpoint of AC Given 2. AM = CM Definition of midpoint 3. M is the midpoint of BD Given 4. BM = DM Definition of midpoint 5. angle AMB = angle CMD Vertical Angles Theorem 6. triangle AMB is congruent to triangle CMD SAS 7. AB = CD CPCTC

Notice how step 6 collects exactly the three pairs above it, and how the angle at step 5 sits between the two pairs of sides — which is why the reason is SAS and not SSS. When a question blanks out one reason, cover the rest and ask only: what does *this line* claim, and what single fact makes it true?

How the questions here ask you to answer

Nothing in this skill asks you to type a proof. The questions come in a few fixed shapes, and each says how to answer.

  • Multiple choice for anything that would be words: which side or angle corresponds, which criterion the givens match, whether there is enough information, which reason fills a blank in a proof, and which extra piece of information would finish a proof. Pick the option; a criterion is written out in full, as SAS (Side-Angle-Side), so the abbreviation and its meaning always travel together.
  • A whole number in degrees when an angle is asked for. Type just the number: 72, not "72 degrees" and not "angle B = 72".
  • A whole number in centimetres when a length is asked for. Again just the number.
  • A whole number for x when two corresponding parts are given as expressions and the question asks for the value of x.

A few conventions in the prompts are worth knowing. A triangle is written "triangle ABC", a side by its two endpoints ("AB", which is the same segment as "BA"), an angle either by its vertex ("angle B") or by three letters with the vertex in the middle ("angle ABC"). Congruence is written out as "is congruent to". From difficulty 4 some questions include a measurement that is perfectly true and completely useless — a perimeter, an area, a single angle of one triangle. Deciding what you do not need is part of the question.

Worked examples

Example 1

Triangle PQR is congruent to triangle ZXY. In triangle PQR, PQ = 9 cm, QR = 14 cm and PR = 11 cm. How long is XY?

  1. Write the two names one above the other and read down the columns: P Q R above Z X Y.
  2. So P matches Z, Q matches X, and R matches Y.
  3. XY is wanted, so run the pairing backwards: X came from Q, and Y came from R.
  4. That means XY corresponds to QR, and QR = 14 cm.
  5. Corresponding sides of congruent triangles are equal, so XY = 14 cm. (Answer: 14)

Example 2

In triangles ABC and DEF you are told AB = DE, BC = EF and angle A = angle D. Is that enough to prove the triangles congruent?

  1. Count the kinds of part: two pairs of sides and one pair of angles, so the letters are S, S and A.
  2. Check the arrangement. AB and BC meet at B, so the included angle for those two sides is angle B.
  3. The angle given is angle A, which is not between them. The pattern is SSA, not SAS.
  4. SSA is not a congruence criterion: with two sides fixed and a non-included angle, the third side can swing into two different positions.
  5. So the answer is: no, SSA is not a congruence criterion. (Had angle B = angle E been given instead, it would have been SAS.)

Example 3

Triangle ABC is congruent to triangle DEF by ASA. Angle B = (5x - 6)° and angle E = (3x + 14)°. How many degrees is angle B?

  1. Angle B and angle E are corresponding parts of congruent triangles, so by CPCTC they are equal.
  2. Set the expressions equal: 5x - 6 = 3x + 14.
  3. Take 3x off both sides: 2x - 6 = 14. Add 6 to both sides: 2x = 20.
  4. Divide by 2: x = 10.
  5. The question asks for angle B, not for x, so substitute: 5(10) - 6 = 44. Angle B = 44°. (Check: 3(10) + 14 = 44 as well.)

Example 4

In the figure, triangle ABC and triangle ADC share the side AC. You are told AB = AD and CB = CD. Which criterion proves the two triangles are congruent, and what does it then give you?

  1. Two matching pairs are given straight away: AB = AD and CB = CD. Both are pairs of sides.
  2. The side AC belongs to both triangles, and any segment equals itself, so AC = AC by the Reflexive Property. That is a third pair, for free.
  3. All three matching pairs are sides, so the criterion is SSS: triangle ABC is congruent to triangle ADC.
  4. By CPCTC every other pair now matches too, so angle ABC = angle ADC and angle BAC = angle DAC.
  5. That is why the shared side is worth hunting for: it turns two given pairs into a finished proof.

Practice problems, with solutions

Three problems of increasing difficulty, each with the full working. In the game these are generated fresh every time; these three are fixed so this page always shows the same ones.

Problem 1

Difficulty 1 of 5

Triangle RST is congruent to triangle UVW. The letters of a congruence statement are matched in the order they are written. Which side of triangle UVW corresponds to side ST?

  1. VW
  2. UW
  3. UV

Answer: A. VW

  1. The statement pairs S with V and T with W.
  2. So the endpoints of ST become V and W.
  3. The corresponding side is VW.

Problem 2

Difficulty 3 of 5

In triangle GHJ and triangle KLM you are told: - HJ = LM - GH = KL - GJ = KM Which congruence criterion do these givens match?

  1. ASA (Angle-Side-Angle)
  2. SSS (Side-Side-Side)
  3. HL (Hypotenuse-Leg)
  4. SAS (Side-Angle-Side)

Answer: B. SSS (Side-Side-Side)

  1. The matching pairs are: GH = KL; HJ = LM; GJ = KM.
  2. That is three pairs of matching sides.
  3. Answer: SSS (Side-Side-Side).

Problem 3

Difficulty 4 of 5

In triangle GHJ and triangle KLM you are told: - GJ = KM - HJ = LM - triangle GHJ has area 24 cm2 Is that enough to prove the two triangles congruent?

  1. Yes, by SSS
  2. No — SSA is not a congruence criterion
  3. No — AAA proves similarity, not congruence
  4. No — two matching parts are not enough

Answer: D. No — two matching parts are not enough

  1. The matching pairs are: HJ = LM; GJ = KM.
  2. Only two matching pairs are listed, and every congruence criterion asks for three.
  3. Answer: No — two matching parts are not enough.

Common mistakes

  • Matching the letters alphabetically instead of by the statement. If it says triangle ABC is congruent to triangle FDE, then AB matches FD and angle B matches angle D — the order written is the order that counts.
  • Accepting SSA as a criterion because "it has three letters". Check where the angle sits: unless it is between the two given sides, the triangle is not pinned down.
  • Treating AAA as congruence. Three matching angles make the triangles the same shape, so they are similar; a photograph and its enlargement have identical angles and very different sizes.
  • Solving for x and stopping there when the question asked for the length or the angle. Once you have x, substitute it back into the expression the question named.
  • Forgetting the shared side. When two triangles are hinged on a common segment, that segment equals itself by the Reflexive Property and gives you a third matching pair without any extra information.
  • Using CPCTC to justify one of the three pairs used in the proof. CPCTC is the payment you receive after the congruence is proved, so it can only appear on a line below it.

What you should be able to do

  • Match corresponding sides and angles of congruent triangles from a congruence statement.
  • Choose the criterion (SSS, SAS, ASA, AAS or HL) that proves two triangles congruent, or decide there is not enough information.
  • Use CPCTC to find an unknown side or angle once triangles are congruent.
  • Supply the missing statement or reason in a two-column congruence proof.

Where this fits in the curriculum

Common Core

  • HSG-CO.B.7

    High school — Use the definition of congruence in terms of rigid motions to show that two triangles are congruent if and only if corresponding pairs of sides and angles are congruent.

  • HSG-CO.B.8

    High school — Explain how the criteria for triangle congruence (ASA, SAS and SSS) follow from the definition of congruence in terms of rigid motions.

  • HSG-CO.C.10

    High school — Prove theorems about triangles.

  • HSG-SRT.B.5

    High school — Use congruence and similarity criteria for triangles to solve problems and to prove relationships in geometric figures.

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