Sec 2 Maths · Congruence and similarity

Congruence tests questions with answers

Practice questions on congruence tests, from the basics up to exam level, each with a full worked solution. Try them on paper first, then open the solution to check your working.

Worked example

In the diagram, the parts marked on triangles $ABC$ and $PQR$ are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.

Step 1 of 2

List the equal parts

\angle ACB = \angle PQR

(both

90^\circ

)

AB = PR

(both

13

cm)

BC = RQ

(both

8

cm)

Try a question: Congruence tests

Free · no sign-up

Answer a real practice question. Get it wrong and MugCorner shows you where it went wrong, with hints and a step-by-step guide.

Questions

Question 1Name the congruence test
In the diagram, the parts marked on triangles $ABC$ and $PQR$ are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.
Show the worked solution
1. Step 1: List the equal parts $AB = PR$ (both $8$ cm) $CA = QP$ (both $7$ cm) $\angle BAC = \angle RPQ$ (both $42^\circ$ )
2. Step 2: Name the testTwo sides and the angle between them are equal: SAS.
Question 2Write the congruence statement
Triangles $ABC$ and $PQR$ are congruent, with the equal parts marked. Complete the statement $\triangle ABC \equiv \triangle \ldots$
Show the worked solution
1. Step 1: Find the part of PQR equal to each marked part of ABC $AB$ ( $10$ cm) matches $\text{RQ}$ $CA$ ( $7$ cm) matches $\text{PR}$ $\angle A$ ( $58^\circ$ ) matches $\angle \text{R}$
2. Step 2: Write the matching vertices in the same order as ABC $A$ matches $R$ , $B$ matches $Q$ , $C$ matches $P$ . $\triangle ABC \equiv \triangle \text{RQP}$
Question 3Find a side using congruence
In the diagram, triangle $ABC$ is congruent to triangle $RPQ$ . Find $x$ .
Show the worked solution
1. Step 1: Match the letters in order $\triangle ABC \equiv \triangle RPQ$ , so $BC$ matches $PQ$ .
2. Step 2: Matching sides of congruent triangles are equal $PQ = BC$ $x = 5$
Question 4Find an angle using congruence
In the diagram, triangle $ABC$ is congruent to triangle $PQR$ . Find $x$ .
Show the worked solution
1. Step 1: Angles in a triangle add up to 180° $\mathstrut \angle C = 180^\circ - 48^\circ - 35^\circ$ $\angle C = 97^\circ$
2. Step 2: Match the letters in order $\triangle ABC \equiv \triangle PQR$ , so $\angle C$ matches $\angle R$ . $x = 97$
Question 5One more condition for congruence
In triangles $ABC$ and $PQR$ , $AB = PQ$ and $\angle ABC = \angle PQR$ . State one more condition that would make triangle $ABC$ congruent to triangle $PQR$ by the SAS test. Choose the condition.
Show the worked solution
1. Step 1: What does the SAS test need?SAS needs two pairs of equal sides and the angles between them.
2. Step 2: What is missing from the given facts?With $BC = QR$ , the triangles are congruent by SAS.

More on congruence and similarity

Notes: Congruence and similarity Worksheet with answers (PDF)

Endless questions on congruence tests

A free account gives you new questions every time, step-by-step help, and tracks what you've mastered.

Start free

Worked example

In the diagram, the parts marked on triangles

ABC

and

PQR

are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.

Step 1 of 2

List the equal parts

\angle ACB = \angle PQR

(both

90^\circ

)

AB = PR

(both

13

cm)

BC = RQ

(both

8

cm)

Questions

Question 1Name the congruence test

In the diagram, the parts marked on triangles $ABC$ and $PQR$ are equal. State the test that shows the two triangles are congruent: SSS, SAS, ASA, AAS or RHS.

Show the worked solution

Step 1: List the equal parts $AB = PR$ (both $8$ cm) $CA = QP$ (both $7$ cm) $\angle BAC = \angle RPQ$ (both $42^\circ$ )
Step 2: Name the testTwo sides and the angle between them are equal: SAS.

Question 2Write the congruence statement

Triangles $ABC$ and $PQR$ are congruent, with the equal parts marked. Complete the statement $\triangle ABC \equiv \triangle \ldots$

Show the worked solution

Step 1: Find the part of PQR equal to each marked part of ABC $AB$ ( $10$ cm) matches $\text{RQ}$ $CA$ ( $7$ cm) matches $\text{PR}$ $\angle A$ ( $58^\circ$ ) matches $\angle \text{R}$
Step 2: Write the matching vertices in the same order as ABC $A$ matches $R$ , $B$ matches $Q$ , $C$ matches $P$ . $\triangle ABC \equiv \triangle \text{RQP}$

Question 3Find a side using congruence

In the diagram, triangle $ABC$ is congruent to triangle $RPQ$ . Find $x$ .

Show the worked solution

Step 1: Match the letters in order $\triangle ABC \equiv \triangle RPQ$ , so $BC$ matches $PQ$ .
Step 2: Matching sides of congruent triangles are equal $PQ = BC$ $x = 5$

Question 4Find an angle using congruence

In the diagram, triangle $ABC$ is congruent to triangle $PQR$ . Find $x$ .

Show the worked solution

Step 1: Angles in a triangle add up to 180° $\mathstrut \angle C = 180^\circ - 48^\circ - 35^\circ$ $\angle C = 97^\circ$
Step 2: Match the letters in order $\triangle ABC \equiv \triangle PQR$ , so $\angle C$ matches $\angle R$ . $x = 97$

Question 5One more condition for congruence

In triangles $ABC$ and $PQR$ , $AB = PQ$ and $\angle ABC = \angle PQR$ . State one more condition that would make triangle $ABC$ congruent to triangle $PQR$ by the SAS test. Choose the condition.

Show the worked solution

Step 1: What does the SAS test need?SAS needs two pairs of equal sides and the angles between them.
Step 2: What is missing from the given facts?With $BC = QR$ , the triangles are congruent by SAS.