Edexcel IGCSE 4MA1 Congruent Triangles: exam questions and answers, notes and worked examples

Congruent Triangles notes

The four conditions

SSS: all three pairs of sides are equal. SAS: two pairs of sides and the angle between them are equal. ASA: two pairs of angles and a pair of matching sides are equal; if you know two angles, the third is fixed by the angle sum, so two angles and any matching side is enough. RHS: both triangles have a right angle, equal hypotenuses and one other pair of equal sides.

What is not enough

Three equal angles give the same shape but not the same size: a triangle with angles ${50}^{\circ }$, ${60}^{\circ }$ and ${70}^{\circ }$ can be any size. Two sides and an angle that is not between them can fit two different triangles, so that does not prove congruence either.

Writing a congruence proof

Give three facts, each with a reason, then name the condition. Triangle $ABC$ is isosceles with $AB=AC$, and $M$ is the midpoint of $BC$. $AB=AC$ (given), $BM=CM$ ($M$ is the midpoint) and $AM$ is common to both triangles, so $ABM$ and $ACM$ are congruent (SSS). That also proves angle $B$ = angle $C$.

Using congruence

Once two triangles are congruent, every matching side and angle is equal. Write the letters in matching order: if triangle $ABC$ is congruent to triangle $PQR$, with angle $A={40}^{\circ }$ and angle $B={65}^{\circ }$, then angle $R$ = angle $C={75}^{\circ }$, and $QR=BC$.