Maths

What are congruent triangles, and how do you prove two triangles are congruent?

Quick answer

Congruent triangles are exactly the same shape and size, with all matching sides and angles equal. You can prove it with SSS, SAS, ASA, AAS or RHS.

Ages 12–16Secondary

In one sentence

Two triangles are congruent when they are exact copies of each other: every side and every angle of one matches a side and an angle of the other.

Explained

Cut a triangle out of paper and use it as a stencil to cut a second one. The two pieces fit perfectly on top of each other, even if you have to turn one round or flip it over first. They are congruent: same shape and same size. Turning (rotation), sliding (translation) and flipping (reflection) never change lengths or angles, which is why transformations like these always give a congruent copy.

scissors
ruler
pencil

Cut two triangles from the same stencil and they are congruent.

We write △ABC ≅ △DEF, which means "triangle ABC is congruent to triangle DEF". The order of the letters tells you which parts match: A with D, B with E, C with F. So AB = DE, BC = EF and AC = DF, and the angles at A and D, B and E, C and F are equal.

A triangle has six parts (three sides, three angles), but you do not need to check all six. Three well-chosen facts are enough. These are the congruence rules:

SSS (side–side–side): all three pairs of sides are equal.
SAS (side–angle–side): two pairs of sides are equal, and the angles between them (the included angles) are equal.
ASA (angle–side–angle): two pairs of angles are equal, and the sides between them are equal.
AAS (angle–angle–side): two pairs of angles are equal, and a pair of matching sides that are not between them. (Two angles fix the third, so this works just like ASA.)
RHS (right angle–hypotenuse–side): both triangles are right-angled, the hypotenuses are equal and one other pair of sides is equal. In the US this is called HL (hypotenuse–leg).

Two combinations do not prove congruence. AAA only proves that triangles are similar: same shape, maybe different size. SSA (two sides and an angle that is not between them) can give two different triangles, so it is not a rule, except in the right-angled case, which is RHS.

Once you know two triangles are congruent, every other matching part is equal too. US textbooks call this CPCTC: corresponding parts of congruent triangles are congruent. It is how congruence is used in proofs, for example to show that the base angles of an isosceles triangle are equal.

Examples

Given in both triangles	Rule	Congruent?
sides 5, 7 and 8 cm	SSS	yes
sides 6 and 9 cm with a 50° angle between them	SAS	yes
angles 40° and 70° with a 10 cm side between them	ASA	yes
angles 30°, 60° and 90°, hypotenuses 10 cm and 12 cm	none	no (similar only)
right angle, hypotenuse 13 cm, one leg 5 cm	RHS	yes (the other leg is 12 cm in both)

A proof. In a kite ABCD, AB = AD and CB = CD. Is △ABC ≅ △ADC? AB = AD (given), CB = CD (given) and AC is a side of both triangles. Three pairs of equal sides: SSS, so yes. By CPCTC, the angles at B and D are equal.

Common mistakes

Using AAA. Equal angles only prove similar triangles. Fix: make sure at least one pair of sides is equal.
Using SAS with the wrong angle. The angle must be between the two sides. If it is not, you have SSA, which is not a rule.
Writing the letters in the wrong order. △ABC ≅ △DEF promises A↔D, B↔E, C↔F. Fix: match the vertices with equal angles before writing the statement.
Missing the "free" facts. A side shared by both triangles, vertically opposite angles and angles made by parallel lines are equal without being told. Look for them first.

Step by step

Mark the diagramPut matching dashes on equal sides and arcs on equal angles from the information given.
Look for free factsA shared side, vertically opposite angles, alternate or corresponding angles on parallel lines, a right angle.
Choose the ruleDo you have SSS, SAS, ASA, AAS or RHS? Check that an SAS angle really is between the two sides.
Write the statementName the triangles with matching vertices in the same order, give the reason for each fact and name the rule.
Use the resultAny other matching sides or angles are now equal too (CPCTC).

Practice questions

Congruent triangles

Try each one, then check your answer.

Two equilateral triangles both have sides of 5 cm. Are they congruent?
Show answer
All three pairs of sides are equal: yes (SSS)
Two triangles both have sides of 4 cm, 6 cm and 7 cm. Congruent? Which rule?
Show answer
Yes, SSS
Two triangles both have sides of 5 cm and 8 cm with a 40° angle between them. Congruent? Which rule?
Show answer
Yes, SAS
Two triangles both have angles of 50° and 60° with a 9 cm side between them. Congruent? Which rule?
Show answer
Yes, ASA
Two triangles both have angles of 35°, 65° and 80°. Nothing is known about the sides. Are they congruent?
Show answer
Not necessarily: AAA only proves they are similar.
Two triangles both have angles of 40° and 75°, and a 6 cm side opposite the 40° angle. Congruent?
Show answer
Yes, AAS (the third angle is 65° in both)
Two right-angled triangles both have a 10 cm hypotenuse and a 6 cm leg. Are they congruent, and how long is the other leg?
Show answer
Yes, RHS. The other leg is √(10² − 6²) = √64 = 8 cm in both.
△ABC ≅ △PQR, AB = 7 cm and angle B = 48°. Find PQ and angle Q.
Show answer
PQ = 7 cm and ∠Q = 48°
△ABC ≅ △XYZ, ∠A = 52° and ∠B = 61°. Find ∠Z.
Show answer
∠C = 180° − 52° − 61° = 67°, and ∠Z matches ∠C: 67°
△KLM ≅ △RST, KL = 3x + 1 and RS = 13. Find x.
Show answer
3x + 1 = 13, so 3x = 12: x = 4
△ABC ≅ △DEF with AB = 5 cm, BC = 6 cm and DF = 4 cm. Find the perimeter of △DEF.
Show answer
DE = 5, EF = 6, DF = 4: 15 cm
Two triangles both have sides of 7 cm and 5 cm and a 30° angle opposite the 5 cm side. Must they be congruent?
Show answer
No: this is SSA. Two different triangles fit these facts (the angle opposite the 7 cm side can be about 44.4° or about 135.6°).
In kite ABCD, AB = AD and CB = CD. Which rule shows △ABC ≅ △ADC?
Show answer
AB = AD, CB = CD and AC is shared: SSS
Segments AB and CD cross at M, with AM = MB and CM = MD. Why is △AMC ≅ △BMD?
Show answer
AM = MB, CM = MD, and ∠AMC = ∠BMD (vertically opposite): SAS
In triangle ABC, AB = AC and M is the midpoint of BC. Prove that ∠B = ∠C.
Show answer
AB = AC, BM = CM and AM is shared, so △ABM ≅ △ACM (SSS). Matching angles are equal: ∠B = ∠C

Word problems

Congruent triangles in real life

Try each one, then check your answer.

Two roof trusses are each made of beams 3 m, 3 m and 5 m long. Are the triangles the same shape and size?
Show answer
Yes, SSS. Three fixed sides fix a triangle completely, which is why triangles keep bridges and roofs rigid.
To find the width AB of a pond, you walk from A to a point C and the same distance again to D, then from B through C to E so that CE = BC. You measure DE = 42 m. How wide is the pond?
Show answer
AC = CD, BC = CE and the angles at C are vertically opposite, so △ABC ≅ △DEC (SAS): AB = DE = 42 m
Two right-triangle supports under a ramp each have a 2.5 m sloping edge (hypotenuse) and a height of 0.7 m. How long is the base of each, and are they congruent?
Show answer
Base = √(2.5² − 0.7²) = √5.76 = 2.4 m; they are congruent by RHS
Two triangular flags are cut with sides of 20 cm and 30 cm meeting at a 40° angle. Will the third sides be the same length?
Show answer
Yes: the flags are congruent by SAS, so all matching parts are equal.
Frame 1 has angles of 50° and 60° with an 8 m beam between them. Frame 2 has angles of 60° and 70° with an 8 m beam between them. Congruent?
Show answer
Both have angles 50°, 60°, 70°, but the 8 m beam is opposite the 70° angle in frame 1 and opposite the 50° angle in frame 2: no

Questions people ask

What does congruent mean in maths?

Congruent means exactly the same shape and size. Congruent triangles can be placed on top of each other so that they match perfectly, after turning, sliding or flipping one of them.

What are the rules for congruent triangles?

There are five: SSS, SAS, ASA, AAS and RHS (called HL in the US). Any one of them is enough to prove that two triangles are congruent.

Why is AAA not a congruence rule?

Because triangles with the same angles can have different sizes. A small and a large equilateral triangle both have three 60° angles, but they are not congruent. AAA proves similarity only.

What is CPCTC?

It stands for "corresponding parts of congruent triangles are congruent". Once you have proved two triangles congruent, you can say that any other pair of matching sides or angles is equal.

What is the difference between congruent and similar triangles?

Congruent triangles have the same shape and size; similar triangles have the same shape but may be different sizes. Every pair of congruent triangles is also similar, with a scale factor of 1.

What is the symbol for congruent?

The symbol is ≅, an equals sign with a wave on top: △ABC ≅ △DEF. Similar uses ~ on its own.

Quick quiz

Congruent triangles

Check what you know with 4 quick questions.

Question 1 of 4

Two triangles have two pairs of equal sides and the angles between those sides are equal. Which rule shows they are congruent?

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What are congruent triangles, and how do you prove two triangles are congruent?

In one sentence

Explained

Examples

Common mistakes

Step by step

Practice questions

Word problems