GCSE Similarity Practice Questions

Write the scale factor as a fraction

Divide the new length by the old length — new over old, in that order.

Simplify the fraction

Cancelling gives the exact scale factor $3$. Leave it as a fraction rather than a rounded decimal.

State the scale factor of the enlargement

So the scale factor from triangle $ABC$ to triangle $PQR$ is $3$.

Work out the scale factor from triangle PQR to triangle ABC

Dividing the new length by the old one gives $k = \frac{3}{2}$.

Multiply the corresponding side by the scale factor

$BC$ corresponds to $QR$, so it is $\frac{3}{2}$ times as long.

Work out the missing length and choose that option

So $BC = 12$ cm — that is the option to pick.

Name the two triangles that are similar

The small triangle sits inside the large one and they share the angle at $A$.

Recall the corresponding angles fact

A straight line crossing two parallel lines makes equal corresponding angles.

Use the parallel line to match a pair of angles

The line $AB$ crosses the two parallel lines $DE$ and $BC$, so angle $ADE$ and angle $ABC$ are corresponding angles and are equal. The same argument on $AC$ gives angle $AED$ = angle $ACB$.

Write down the angles that gives you

So angle $ADE = 48^\circ$ and angle $AED = 67^\circ$.

Rule out wrong option number 1

Angle $AED$ is really $67^\circ$, not $48^\circ$, so this option is false.

State the option you have chosen

So angle $AED$ is $67^\circ$.

Name the two triangles that are similar

The small triangle sits inside the large one and they share the angle at $A$.

Recall the corresponding angles fact

A straight line crossing two parallel lines makes equal corresponding angles.

Work out the angle at A

The angles of triangle $ABC$ add to $180^\circ$, so the angle at $A$ is $65^\circ$. It belongs to both triangles.

Note that the two triangles have the same three angles

That is why they are similar: the shape is identical, only the size differs.

Use the parallel line to match a pair of angles

The line $AB$ crosses the two parallel lines $DE$ and $BC$, so angle $ADE$ and angle $ABC$ are corresponding angles and are equal. The same argument on $AC$ gives angle $AED$ = angle $ACB$.

Write down the angles that gives you

So angle $ADE = 52^\circ$ and angle $AED = 63^\circ$.

Rule out wrong option number 1

Angle $AED$ is really $63^\circ$, not $52^\circ$, so this option is false.

Rule out wrong option number 2

Angle $AED$ is really $63^\circ$, not $117^\circ$, so this option is false.

Rule out wrong option number 3

Angle $ADE$ is really $52^\circ$, not $63^\circ$, so this option is false.

State the option you have chosen

So angle $AED$ is $63^\circ$.

Recall the angle test for similar triangles

If two pairs of corresponding angles are equal the triangles are similar. No side lengths are needed at all.

Recall the angle sum of a triangle

This lets you fill in the third angle of each triangle before comparing them.

Note what the question gives you

Only angles are given, so any reason that talks about the ratio of sides cannot be used here.

Plan the comparison

Work out the missing angle in each triangle, then compare the two sets of three.

Note that the correspondence matters

The letters pair the vertices, so compare angle $BAC$ with angle $QPR$, not with angle $PQR$.

Work out the third angle of triangle ABC

The angles of a triangle add to $180^\circ$, so angle $ACB = 65^\circ$.

Work out the third angle of triangle PQR

In the same way angle $PRQ = 55^\circ$.

Compare the two sets of three angles

The two triangles do not have the same three angles, so they are not the same shape.

State the correct conclusion

The third angles differ, so the triangles cannot be similar.

Rule out the SSS reason

SSS needs the three pairs of sides to be in the same ratio, but the question gives no side lengths at all, so this reason cannot be used.

Rule out the SAS reason

SAS needs two pairs of sides in the same ratio, and again there are no side lengths in the question, so this reason is not available either.

Rule out the congruence claim

Doubling every side of a triangle keeps all three angles the same but changes the triangle, so equal angles alone can never prove congruence.

Rule out the remaining option

The claim that two pairs of angles are equal is false here: angle $ABC = 60^\circ$ but angle $PQR = 70^\circ$.

Note that similar shapes can be any size

Similarity fixes the shape, not the size — that is the whole point of a scale factor.

State the option you have chosen

So the triangles are not similar, because their third angles are $65^\circ$ and $55^\circ$, which are different.