Similar Triangles: GCSE Maths Guide

What does similar mean in geometry?

Two shapes are similar if one is an enlargement of the other. For triangles, similarity means:

• All three corresponding angles are equal.
• All three pairs of corresponding sides are in the same ratio (the scale factor).

Note: congruent triangles are also equal in size (scale factor = 1). Similar includes congruent as a special case, but at GCSE the two terms are kept separate.

Notation: When writing that triangles ABC and PQR are similar, the letter order matters — A corresponds to P, B to Q, and C to R. Write triangle ABC ~ triangle PQR (~ means "is similar to").

What are the three conditions for similar triangles?

You only need to verify one of these three conditions to establish similarity:

AA is the most common condition at GCSE. Because the angles in any triangle sum to 180°, two matching angles guarantee the third automatically.

How do you prove two triangles are similar using AA?

1. Identify two angles in one triangle and their corresponding angles in the other.
2. State each pair with a reason (e.g., "vertically opposite angles", "alternate angles", "angles on a straight line", or "given").
3. Conclude: "Since two angles in triangle ABC equal the corresponding angles in triangle PQR, the triangles are similar by AA."

Worked example: In a diagram, triangles ABD and CBE share vertex B. Angle ADB = angle CEB = 90°, and angle B is common to both.

1. Angle ADB = angle CEB = 90° (given).
2. Angle ABD = angle CBE (same angle — the angle at B).
3. Two angles match → triangles ABD and CBE are similar by AA.

How do you find a missing side using similarity?

Once you know two triangles are similar, set up a ratio using corresponding sides.

Method:

1. Identify which sides in each triangle correspond to each other.
2. Write the ratio: (side in larger triangle) / (side in smaller triangle) = scale factor k.
3. Use k to find any missing side: missing side = known side × k.

Worked example: Triangle PQR ~ triangle XYZ. PQ = 6 cm, QR = 9 cm, PR = 12 cm. XY = 4 cm. Find YZ and XZ.

1. Corresponding sides: PQ ↔ XY, QR ↔ YZ, PR ↔ XZ.
2. Scale factor: k = XY / PQ = 4 / 6 = 2/3.
3. YZ = QR × (2/3) = 9 × (2/3) = 6 cm.
4. XZ = PR × (2/3) = 12 × (2/3) = 8 cm.

Check: Verify all three ratios are equal: 4/6 = 6/9 = 8/12 = 2/3. ✓

How do you handle similarity in a diagram with overlapping triangles?

A common GCSE setup places a smaller triangle inside a larger one, sharing a vertex and a pair of parallel sides.

Worked example: In triangle ADE, B lies on AD and C lies on AE such that BC is parallel to DE. AB = 3 cm, BD = 6 cm, BC = 5 cm. Find DE.

1. Because BC ∥ DE, alternate angles give: angle ABC = angle ADE and angle ACB = angle AED.
2. By AA, triangle ABC ~ triangle ADE.
3. AD = AB + BD = 3 + 6 = 9 cm.
4. Scale factor: AD/AB = 9/3 = 3.
5. DE = BC × 3 = 5 × 3 = 15 cm.

What is the difference between similar and congruent triangles?

When writing a similarity proof, always state the specific condition used (AA, SAS or SSS). "Same angles" is not enough on its own — you must link it to the condition name.

Frequently asked questions

Why do only two equal angles prove similarity (AA), not three?

Because if you know two angles of a triangle, the third is determined by the angle sum rule (angles in a triangle sum to 180°). Finding two matching angles is therefore equivalent to finding all three — a third check would be redundant.

What if I am not told which angles are equal — how do I find them?

Look for angles that are equal for geometric reasons: vertically opposite angles, alternate angles (parallel lines), corresponding angles (parallel lines), angles in the same segment of a circle, or angles that are simply shared (the same angle appears in both triangles because the triangles overlap). State each reason explicitly in your proof.

How do I find the scale factor if both triangles are in the same diagram?

Identify one pair of corresponding sides where you know both lengths, then divide the larger by the smaller (or smaller by larger, depending on the direction you want to work in). Be consistent: always divide image by object, or always object by image, and apply the same factor to every side.

Can similar triangles have the same area?

Only if the scale factor is exactly 1 — in which case they are congruent, not just similar. For any scale factor k ≠ 1, the areas differ: if the sides are in ratio k, the areas are in ratio k². For example, if two similar triangles have sides in ratio 2:3, their areas are in ratio 4:9.

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