Similar and congruent shapes explained

Two triangles are congruent if they match in one of these ways:

AAA isn't enough: it proves the triangles are similar, not congruent. SSA isn't a condition either, except as RHS.

If shape B is shape A enlarged by scale factor $k$:

• lengths multiply by $k$;
• areas multiply by $k^2$;
• volumes multiply by $k^3$.

Example: two similar cylinders have heights 4 cm and 6 cm, so $k = 1.5$. The area scale factor is $1.5^2 = 2.25$ and the volume scale factor is $1.5^3 = 3.375$. If the small cylinder holds 80 cm³, the large one holds $80 \times 3.375 = 270$ cm³.

1. Match the corresponding sides (often by matching equal angles).
2. Find the scale factor: a length in the big shape ÷ the matching length in the small one.
3. Multiply or divide to find the missing length.

Redrawing the two shapes the same way round makes matching sides much easier to see.

If you know the area ratio, square-root it to get $k$; if you know the volume ratio, cube-root it. Two similar shapes with areas 20 cm² and 45 cm² have an area scale factor of 2.25, so $k = \sqrt{2.25} = 1.5$.

A congruence proof states the matching parts with a reason for each, then the condition:

"$AB = DE$ (given), $\angle ABC = \angle DEF$ (given), $BC = EF$ (given). So triangles $ABC$ and $DEF$ are congruent (SAS)."

Every equal side or angle needs a reason, such as "given", "common side" or "vertically opposite angles".

• Using the length scale factor for areas or volumes.
• Matching the wrong sides.
• Claiming AAA proves congruence.

Practise with our similar shapes questions, and see how enlargements create similar shapes in transformations.

When a line is drawn parallel to one side of a triangle, it cuts off a smaller triangle that is similar to the original: the angles are equal (corresponding angles on parallel lines, and the shared angle). Many exam questions hide similar triangles this way, often in an "hourglass" shape where two triangles meet at a point.

1. Two similar triangles have corresponding sides of 5 cm and 15 cm. The smaller has an area of 12 cm². Find the larger area.
2. Two similar solids have volumes in the ratio 8 : 27. What is the ratio of their lengths?
3. Two similar bottles are 10 cm and 15 cm tall. The smaller holds 400 ml. How much does the larger hold?

Answers: 1. $k = 3$, so area $= 12 \times 9 = 108$ cm². 2. Cube-root both: $2 : 3$. 3. $k = 1.5$, so $400 \times 1.5^3 = 1350$ ml.

State the scale factor as a line of working, and say whether it's for lengths, areas or volumes. Many marks are lost by using the length scale factor in an area question.