Triangle Congruence Criteria for the TMUA

Understanding Geometric Congruence

In geometry, two figures are described as congruent if they are exactly the same shape and size. If two triangles are congruent, all their corresponding side lengths are equal and all their corresponding interior angles are equal. While a triangle has six primary measurements (three sides and three angles), you do not need to know all six to prove congruence. Instead, specific combinations of three measurements are sufficient to 'lock' the triangle into a unique shape.

The Four Criteria for Congruence

You must be able to recognise and use the following four criteria in TMUA questions:

1. SSS (Side-Side-Side): If the three sides of one triangle are equal to the three sides of another triangle, the triangles are congruent. This is because a triangle with fixed side lengths has no 'flexibility', the interior angles are entirely determined by the side lengths via the Cosine Rule, $a^2 = b^2 + c^2 - 2bc \cos A$.

2. SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent. The term 'included' is vital: the angle must be the one located between the two known sides. If the angle is not between the sides, the triangle is not necessarily unique (see the 'SSA' caution below).

3. ASA (Angle-Side-Angle): If two angles and the side between them are equal to the corresponding parts of another triangle, they are congruent. Note that since the angles of a triangle always sum to $180^\circ$, knowing any two angles automatically gives the third. Therefore, AAS (Angle-Angle-Side) is also a valid proof of congruence, provided the side is in the same relative position in both triangles.

4. RHS (Right-angle, Hypotenuse, Side): This is a special case for right-angled triangles. If two right-angled triangles have equal hypotenuses and one other equal side, they are congruent. According to Pythagoras' Theorem, $a^2 + b^2 = c^2$, if the hypotenuse and one side are fixed, the third side must also be equal, effectively satisfying the SSS criterion.

Identifying Non-Congruent Triangles

It is equally important to know which combinations do not prove congruence:

• AAA (Angle-Angle-Angle): Having three equal angles proves that triangles are similar (the same shape), but not necessarily congruent (the same size). One triangle could be a scaled-up version of the other.
• SSA (Side-Side-Angle): If you know two sides and an angle that is not included between them, there may be two different possible triangles. This is known as the 'ambiguous case' of the Sine Rule, where the unknown side can take two different lengths while keeping the given side-side-angle measurements constant.

Worked Example: Applying Criteria in Proofs

Consider a circle with centre $O$. Let $AB$ be a chord, and let $M$ be the midpoint of $AB$. Prove that the line $OM$ is perpendicular to $AB$.

1. Draw the radii $OA$ and $OB$.
2. In triangles $\triangle OAM$ and $\triangle OBM$, we have:
   • $OA = OB$ (both are radii of the same circle).
   • $AM = MB$ (given that $M$ is the midpoint).
   • $OM = OM$ (shared common side).
3. Therefore, $\triangle OAM \cong \triangle OBM$ by the SSS criterion.
4. Because the triangles are congruent, the corresponding angles $\angle OMA$ and $\angle OMB$ must be equal.
5. Since $AMB$ is a straight line, $\angle OMA + \angle OMB = 180^\circ$.
6. As the angles are equal and sum to $180^\circ$, each must be $90^\circ$, proving that $OM$ is perpendicular to $AB$.