GCSE · Congruence & similarity

Congruent triangles & proofExam-style questions

The four congruence conditions SSS, SAS, ASA/AAS and RHS, why SSA is not one of them, and how to write a congruence proof that earns every mark.

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15 questions · GCSE (Higher) level · mark scheme and worked solution on each · no sign-up to try

DifficultyAll Level 1 Level 2 Level 3 Level 4 Level 5

1
Triangle $ABC$ has $AB = 8$ cm and $BC = 5$ cm, and the angle at $A$ (which is not between those two sides) is $30°$ .

(a) Show that there are two different triangles that fit this description, by working out the two possible sizes of angle $ACB$ . Give your answers correct to 1 decimal place.

(b) Explain what this shows about "SSA" as a condition for congruence.
Level 5 · 5 marksTry it
2
$ABC$ is an isosceles triangle with $AB = AC$ . $M$ is the midpoint of $BC$ .

(a) Prove that triangle $ABM$ is congruent to triangle $ACM$ .

(b) Hence prove that $AM$ is perpendicular to $BC$ .
Level 5 · 6 marksTry it
3
$A$ and $B$ are points on a circle with centre $O$ . $M$ is the midpoint of the chord $AB$ .(a) Prove that triangle $OAM$ is congruent to triangle $OBM$ .

(b) Hence explain why $OM$ is perpendicular to $AB$ .
Level 5 · 4 marksTry it
4
$ABC$ is an isosceles triangle with $AB = AC$ .

$D$ is a point on $AB$ and $E$ is a point on $AC$ such that $BD = CE$ .

(a) Prove that triangle $DBC$ is congruent to triangle $ECB$ .

(b) Hence show that $DC = EB$ .
Level 5 · 4 marksTry it