Congruence

Geometry and measures

Questions marked Higher are aimed at the Higher tier.

These are original practice questions, not a past paper. This site is independent and not affiliated with AQA or Pearson.

State four conditions for two triangles to be congruent.

2 marks
One triangle has sides $5$ cm, $5$ cm and $6$ cm. Another has sides $5$ cm, $6$ cm and $5$ cm. Are the triangles congruent?

2 marks
In triangles $ABC$ and $DEF$, $AB = DE$, $AC = DF$ and angle $A$ equals angle $D$. Angle $A$ is between sides $AB$ and $AC$, and angle $D$ is between sides $DE$ and $DF$. Are the triangles congruent?

3 marks
Two right-angled triangles each have hypotenuse $13$ cm and a leg of $5$ cm. Find the third side of each triangle, and name the congruence condition.

3 marks
$ABCD$ is a rectangle. Prove that triangles $ABC$ and $CDA$ are congruent.

4 marks
Triangles $ABC$ and $DEC$ meet at $C$, with $AB$ parallel to $DE$. Angle $BAC = $ angle $EDC$, and $AC = DC$. Angle $ACB$ and angle $DCE$ are vertically opposite. Prove that the triangles are congruent.

4 marks
$PQRS$ is a parallelogram. Prove that triangles $PQS$ and $RSQ$ are congruent. Higher

5 marks
Triangle $ABC$ is right-angled at $C$, with $AC = 8$ cm and $AB = 10$ cm. Triangle $DEF$ is right-angled at $F$, with $DF = 8$ cm and $DE = 10$ cm. Prove that the triangles are congruent and find $BC$ and $EF$. Higher

6 marks

Mark scheme

Answer. SSS, SAS, ASA and RHS
- Three sides; two sides and the included angle; two angles and a corresponding side.
- Right angle, hypotenuse and another side.
Answer. Yes, by SSS
- The three lengths are the same.
- SSS
Answer. Yes, by SAS
- Two pairs of sides are equal.
- The equal angles are the included angles, so SAS applies.
Answer. Both third sides are $12$ cm. RHS.
- $\sqrt{13^{2} - 5^{2}} = 12$
- The hypotenuse and a corresponding side of a right-angled triangle match, so RHS applies.
Answer. SSS, using the opposite sides and the shared diagonal
- $AB = CD$ and $BC = DA$, opposite sides of a rectangle.
- $AC$ is common.
- The triangles are congruent by SSS.
Answer. ASA
- The equal angles at $A$ and $D$ are alternate angles between the parallel lines.
- The vertically opposite angles at $C$ are equal, and $AC = DC$.
- Two angles and the corresponding side give ASA.
Answer. SSS
- $PQ = RS$ and $PS = RQ$, opposite sides of a parallelogram.
- The diagonal $QS$ is common.
- The triangles are congruent by SSS.
Answer. RHS. Both remaining sides are $6$ cm.
- Each triangle has a right angle, hypotenuse $10$ cm and another side $8$ cm.
- RHS, so the triangles are congruent.
- $BC = \sqrt{10^{2} - 8^{2}} = 6$ cm, and $EF = 6$ cm as well.