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SPI Mathematics · Triangles

Congruence of Triangles

Two triangles are congruent when one fits exactly on the other, and three well-chosen equal parts (SSS, SAS, ASA, AAS or RHS) are enough to prove it.

All Triangles notes

Why this matters

Eleven questions in the bank, nine of them EASY. None asks for a long proof: they ask which rule fits, which part is still needed, or which way round the letters go. The two MODERATE ones both turn on order, the included angle in SAS and the matching of vertices.

Concept 1 of 3: Congruent triangles and the order of the letters

Congruent triangles are exact copies: same three sides and same three angles. The statement △ABC≅△PQR\triangle ABC \cong \triangle PQR also tells you which corner sits on which: first on first, second on second, third on third.

Definition

  • Congruent means all six parts match: three sides and three angles.
  • In △ABC≅△PQR\triangle ABC \cong \triangle PQR, A goes to P, B to Q and C to R.
  • So the matching sides are AB and PQ, BC and QR, CA and RP, and the matching angles are A and P, B and Q, C and R.
  • To find the order from given sides, look for the corner two equal sides share: that corner matches the corner the other two sides share.

Matching parts

△ABC≅△PQR⇒AB=PQ, BC=QR, CA=RP, ∠A=∠P, ∠B=∠Q, ∠C=∠R\triangle ABC \cong \triangle PQR \Rightarrow AB = PQ,\ BC = QR,\ CA = RP,\ \angle A = \angle P,\ \angle B = \angle Q,\ \angle C = \angle R

Worked example

If △DEF≅△LMN\triangle DEF \cong \triangle LMN, which side equals EF and which angle equals ∠F\angle F?
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 1 · Triangles · Congruence of Triangles

If AB = QR, BC= PR and CA = PQ, then

Matching letters in alphabetical order

△ABC≅△PQR\triangle ABC \cong \triangle PQR is only right if A really matches P. When AB = QR, B and R are not paired by alphabet but by the sides they share. Work out each corner before writing the statement.

Equal angles alone are not congruence

Two triangles with the same three angles can be different sizes. They are similar, not congruent, until a pair of matching sides is also equal.
Drill 4 more on congruent triangles and the order of the letters

Concept 2 of 3: The congruence rules SSS, SAS, ASA, AAS and RHS

You never need all six parts. Three of the right kind lock the triangle's shape and size, so the other three must match too. The rule's name lists those three parts in order round the triangle.

Definition

Two triangles are congruent when one of these holds:

  • SSS: three sides equal.
  • SAS: two sides and the angle between them.
  • ASA: two angles and the side between them.
  • AAS: two angles and a side that is not between them.
  • RHS: a right angle, the hypotenuse and one other side.

AAA and SSA are not congruence rules.

RuleParts that must be equalNote
SSSAll three pairs of sidesNo angle is needed
SASTwo sides and the angle they makeThe angle must sit between the two sides
ASATwo angles and the side joining their cornersThe side must sit between the two angles
AASTwo angles and a side not between themWorks because the third angles are then equal too
RHSRight angle, hypotenuse, one other sideOnly for right triangles
AAAThree anglesNot a congruence rule
Equal angles give similar triangles, which may differ in size.
SSATwo sides and an angle not between themNot a congruence rule
The right-angled case of this pattern is RHS, which does work.
The letters name the parts in order round the triangle: S for a side, A for an angle.
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionEasy

Example 2 · Triangles · Congruence of Triangles

In triangles ABC and PQR, ∠\angle B = 90∘90^\circ, AC = 8 cm, AB = 3 cm, ∠\angle P = 90∘90^\circ, PR = 3 cm, QR = 8 cm. By which congruence rule the triangles are congruent?

Calling RHS when the side given is not the hypotenuse

RHS needs the HYPOTENUSE of each right triangle to be equal, plus one more side. Two equal legs with the right angle between them is SAS, not RHS.
Drill 3 more on the congruence rules sss, sas, asa, aas and rhs

Concept 3 of 3: The included angle and the included side

An angle sits at a corner, between the two sides that meet there. A side joins two corners, so it sits between the angles at its two ends. SAS and ASA only work with these 'in between' parts.

Definition

  • The angle included between sides AB and BC is the angle at their shared letter: ∠B\angle B.
  • The side included between ∠P\angle P and ∠Q\angle Q is the side joining those letters: PQ.
  • To use SAS, find the corner the two given sides share; to use ASA, join the two given corners.

Worked example

You want to prove △XYZ≅△LMN\triangle XYZ \cong \triangle LMN by ASA, and you know ∠X=∠L\angle X = \angle L and ∠Y=∠M\angle Y = \angle M. Which pair of sides must also be equal?
Practice this conceptself-check · 4 quick reps

The same idea in a practice question:

Practice questionModerate

Example 3 · Triangles · Congruence of Triangles

In△\triangle ABC and △\triangle PQR, AB = PR and ∠A\angle A = ∠P\angle P . Then, the two triangles will be congruent by SAS axiom if:

An angle that is not between the sides

Two equal sides and an equal angle at the wrong corner is SSA, and it does not prove congruence. Check that the angle's letter is the letter the two sides share.
Drill 1 more on the included angle and the included side

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (1)

  • Congruent triangles and the order of the letters

    Matching parts

    △ABC≅△PQR⇒AB=PQ, BC=QR, CA=RP, ∠A=∠P, ∠B=∠Q, ∠C=∠R\triangle ABC \cong \triangle PQR \Rightarrow AB = PQ,\ BC = QR,\ CA = RP,\ \angle A = \angle P,\ \angle B = \angle Q,\ \angle C = \angle R

Reference tables (1)

The congruence rules SSS, SAS, ASA, AAS and RHS7 rows
RuleParts that must be equalNote
SSSAll three pairs of sidesNo angle is needed
SASTwo sides and the angle they makeThe angle must sit between the two sides
ASATwo angles and the side joining their cornersThe side must sit between the two angles
AASTwo angles and a side not between themWorks because the third angles are then equal too
RHSRight angle, hypotenuse, one other sideOnly for right triangles
AAAThree anglesNot a congruence rule
Equal angles give similar triangles, which may differ in size.
SSATwo sides and an angle not between themNot a congruence rule
The right-angled case of this pattern is RHS, which does work.
The letters name the parts in order round the triangle: S for a side, A for an angle.

Watch out for (4)

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