Congruence, Similarity & Geometrical Proof Practice Questions

Write down the congruence criterion that shows triangle $PQR$ is congruent to triangle $XYZ$.

In triangle $PQR$, $PR = 8.3$ cm and $\angle QPR = 48^\circ$.

Write down

(i) the length of $XZ$,

(ii) the size of $\angle YZX$.

Triangle $LMN$ has $LM = 7$ cm, $MN = 9$ cm and $\angle MNL = 62^\circ$.

Explain why triangle $LMN$ is not necessarily congruent to triangle $PQR$.

Work out the length of $BC$.

$AC = 12$ cm.

Work out the length of $EC$.

Prove that triangle $ABM$ is congruent to triangle $ACM$.

Use your answer to part (a) to explain why $AM$ bisects $BC$ and $AM$ bisects $\angle BAC$.

$ACE$ and $BCD$ are straight lines. $AB$ is parallel to $DE$.

Prove that triangle $ABC$ and triangle $EDC$ are similar.

You must give reasons for each stage of your working.

Some water is poured into the funnel so that the depth of the water is $12$ cm.

Work out the radius of the circular surface of the water.

More water is poured in so that the circular surface of the water has diameter $9$ cm.

Work out the depth of the water now.

Prove that triangle $ADE$ is similar to triangle $ACB$.

You must give reasons for your answer.

Work out the length of $EC$.

Prove that triangle $ABD$ is similar to triangle $CBA$.

You must give reasons for each stage of your working.

Work out the length of $AD$.

Prove that triangle $AXB$ is similar to triangle $CXD$.

You must give reasons for your answer.

$AC = 16$ cm.

Work out the length of $AX$.

The perpendicular distance between $AB$ and $DC$ is $12$ cm.

Work out the perpendicular distance from $X$ to $AB$.

Prove that triangle $ADP$ is congruent to triangle $BCP$.

Work out the size of $\angle DPC$.

You must give reasons for each stage of your working.