Congruence & Similarity

SPEC G5, G6, G19 · GEOMETRY & MEASURES

Criteria for congruent triangles, identifying similar shapes, and using the scale factor to find missing lengths

What you will learn

By the end of this lesson you will:

State the four congruence criteria for triangles: SSS, SAS, ASA, RHS

Identify a pair of congruent triangles and give the correct reason

Use congruence to prove simple results — including the base angles of an isosceles triangle are equal

Explain when two shapes are similar and state the similarity criteria

Find missing lengths in similar figures using the scale factor k

Preview how k extends to area and volume Higher preview — covered in full in Lesson 4.08

Why this matters

Recent examiner reports highlight "identifying the reason for congruence" as a topic where well-prepared students score full marks — but those who write vague reasons lose them all. This lesson locks that down.

The Big Idea: Congruent vs Similar

CONGRUENT

Same shape AND same size.

You could pick one up and place it exactly on top of the other (after rotating or flipping if needed).

SIMILAR

Same shape, different size.

One is an enlargement of the other. All angles match; all sides grow (or shrink) by the same factor.

Insight

Congruent is just a special case of similar — where the scale factor happens to be 1.

Definition: Congruent

CONGRUENT

Two shapes are congruent if all corresponding sides are equal AND all corresponding angles are equal.

Practical test

If you could cut one shape out and place it exactly on top of the other — possibly after rotating it or flipping it over — they are congruent.

Two shapes, many ways they can be congruent

Congruent shapes can be in different orientations. A triangle pointing left and a triangle pointing right can still be congruent. Always compare corresponding parts, not the "look".

The Four Congruence Criteria

For triangles, you only need to check one of these to prove congruence. These four are the official AQA spec criteria (G5).

CodeWhat it stands forWhat you need to show

SSSSide Side SideAll three pairs of sides are equal.

SASSide Angle SideTwo pairs of sides equal AND the angle between them (the included angle) is equal.

ASAAngle Side AngleTwo pairs of angles equal AND the side between them is equal.

RHSRight angle Hypotenuse SideRight-angled triangles only: right angle, equal hypotenuse, and one other pair of sides equal.

About AAS

Some textbooks list a fifth criterion AAS (Angle-Angle-Side). If two pairs of angles match, the third pair must match too (angles sum to 180°), so AAS reduces to ASA. AQA's spec lists the four above — use those names in your answer.

Spotting which criterion to use

Read what information is given, then pick the matching criterion.

You are told...Use...

All three sides of each triangleSSS

Two sides and the angle between themSAS

Two angles and the side between themASA

Both triangles have a right angle, equal hypotenuse, and one other equal sideRHS

⚠️ Watch the "included" word.SAS needs the angle between the two sides. If the equal angle is not between them, SAS does not apply — and the triangles may not even be congruent. Same for ASA: the side must be between the two angles.

Worked Example 1

01Identify the Congruent Pair 2 marks

The diagram shows three triangles with the side lengths labelled.

Which two triangles are congruent? Give the reason.

M

List the three sides of each triangle and look for a matching set. A has sides 5, 7, 8 cm. B has sides 6, 7, 9 cm. C has sides 5, 7, 8 cm.

C

A: {5, 7, 8} ✓ B: {6, 7, 9} ✗ C: {5, 7, 8} ✓. A and C have identical side sets. All three pairs of sides match.

E

Triangles A and C are congruent by SSS. Orientation does not affect congruence — C is just a reflected version of A.

Examiner Alert: Name the Reason

⚠️ EXAMINER ALERT — "they look the same" scores zero.When asked "give the reason", examiners want one of the four codes (SSS, SAS, ASA, RHS) — nothing else scores.

Don't writeWrite instead

"They look identical"SSS (or the correct code)

"Same triangle"SSS — all three sides equal

"Same angles and sides"ASA / SAS / RHS as appropriate

"Congruent because one flipped over"Orientation is irrelevant — state the criterion used.

Exam tip

When you write the code, briefly state which parts match. Example: "SSS — the 5, 7 and 8 cm sides of A match the 5, 7 and 8 cm sides of C." This removes any ambiguity and usually secures both marks.

Definition: Similar

SIMILAR

Two shapes are similar if all corresponding angles are equal AND all corresponding sides are in the same ratio.

Another way to say it

One shape is an enlargement of the other. Every length is multiplied by the same scale factor k. The angles don't change.

Link to Lesson 4.06

Enlargements produce similar shapes. An enlargement with scale factor 3 makes a new shape that is similar to the original, with all sides 3× as long.

Similarity Criteria (for Triangles)

You only need to show one of these to prove two triangles are similar.

CodeWhat to show

AAAAll three pairs of corresponding angles are equal. (In practice, showing two pairs equal is enough — the third is forced.)

Sides in ratioAll three pairs of corresponding sides are in the same ratio .

SAS-similarTwo pairs of sides in the same ratio AND the included angle equal.

Most common proof route

At GCSE, AAA is by far the most common route — usually via angle facts (shared angle, parallel-line angles, or angles in the same segment).

⚠️ AAA works for triangles only. For other shapes (quadrilaterals, polygons, etc.) equal angles alone are not enough — you must also check that corresponding sides are in the same ratio. A square and a non-square rectangle both have four 90° angles but are not similar. The definition on the previous slide (both conditions) is the safe rule for any shape.

The Scale Factor k

SCALE FACTOR

The length scale factork between two similar shapes is the ratio of one pair of corresponding sides.

k = new lengthoriginal length

k > 1

The new shape is larger (an enlargement).

0 < k < 1

The new shape is smaller (a reduction).

The golden rule

Once you know k, every corresponding pair of sides is related by the same k. Multiply by k to go from small to large; divide by k to go from large to small.

Worked Example 2

02Missing Sides in Similar Triangles 3 marks

Two triangles are similar. The smaller triangle has sides of length 4 cm, 6 cm and 8 cm. In the larger triangle, the side corresponding to the 4 cm side is 10 cm.

Find the other two sides of the larger triangle.

M

Use the pair that's fully given (4 cm ↔ 10 cm) to find the scale factor, then apply it to the other two sides, matching each with its corresponding partner.

C

k = 10 ÷ 4 = 2.5. Side corresponding to 6 cm: 6 × 2.5 = 15 cm. Side corresponding to 8 cm: 8 × 2.5 = 20 cm.

E

The larger triangle has sides 10 cm, 15 cm and 20 cm. Check: the three ratios 10:4, 15:6, 20:8 all simplify to 5:2 — every corresponding pair shares the same ratio, confirming similarity ✓.

Matching Corresponding Sides

The trickiest part is knowing which side in one triangle corresponds to which side in the other — especially when they are different sizes. The two triangles below are similar (both are 3-4-5 right triangles; the larger is a 6-8-10). Use the matching marks to spot corresponding angles — then pair up the sides opposite them.

How to match them

Same angle mark = corresponding angles. Here the right-angle squares pair up, single-arc pairs up, double-arc pairs up.

The side opposite each matching angle is the corresponding side. So 5 cm (opposite right angle on the left) corresponds to 10 cm (opposite right angle on the right). Likewise 3 ↔ 6 and 4 ↔ 8.

Or: read the vertex-label order. If the question says ABC is similar to DEF, then A↔D, B↔E, C↔F, and AB corresponds to DE.

Ratio check

With correct pairing, all ratios of corresponding sides should be equal: 6:3 = 8:4 = 10:5 = 2. k = 2, confirming similarity. If one ratio doesn't match, you've paired the wrong sides — go back to the angles.

Examiner Alert: Incorrect Pairing

⚠️ EXAMINER ALERT — mismatched corresponding sides.The most common mistake in similar-shape questions is pairing the wrong sides when computing the scale factor. A single wrong pairing wastes all 3 marks.

Weak studentStrong student

Sees sides 4, 6, 8 and 10 — assumes the 10 must correspond to 6 or 8.Reads the question: "10 cm corresponds to 4 cm" — uses that pair.

Uses k = 10 ÷ 6 1.67 — wrong SF ripples into every later calculation.Uses k = 10 ÷ 4 = 2.5 — correct, all subsequent sides correct.

Never checks if the other sides look sensible.Sanity-checks: do the final lengths look proportional to the original?

Process

(1) Name the corresponding pair (by angle or vertex order). (2) Compute k. (3) Apply k to every other length. (4) Ratio-check the final answer.

Similar Triangles from Parallel Lines

A classic exam setup: a triangle with a line drawn parallel to one of its sides. This always creates two similar triangles.

Why they're similar — the proof (AAA)

Angle A is shared by both ADE and ABC.

Angle ADE = ABC (corresponding angles, DE BC).

Angle AED = ACB (corresponding angles, DE BC).

All three angles match → ADE is similar to ABC.

Pattern spotting

Whenever you see parallel lines inside a triangle, expect similar triangles. The smaller one sits inside the larger, sharing the apex.

Using Congruence in Proof

Spec point G6 asks you to use congruence to derive results and obtain simple proofs. The classic example: proving the base angles of an isosceles triangle are equal.

03Isosceles Base Angles 3 marks

Triangle ABC has AB = AC (an isosceles triangle). M is the foot of the perpendicular from A to BC. Prove that ABC = ACB.

M

Compare ABM and ACM. Find three matching pieces that satisfy one of the congruence criteria, then deduce the equal angles as corresponding parts.

C

In ABM and ACM:

AB = AC (given — isosceles)

AM = AM (common side)

AMB = AMC = 90° (AM is perpendicular to BC)

So ABM ACM by RHS (right angle, equal hypotenuse AB = AC, common side AM).

E

Corresponding angles in congruent triangles are equal, so ABM = ACM, i.e. ABC = ACB. The base angles of an isosceles triangle are equal. ∎

Exam structure for congruence proofs

(1) State the two triangles being compared. (2) List three matching pieces with reasons (given, common, construction, angle fact). (3) Name the criterion (SSS / SAS / ASA / RHS). (4) Conclude by stating what follows from the congruence.

Side-by-Side: Congruent vs Similar

PropertyCongruentSimilar

Corresponding anglesAll equalAll equal

Corresponding sidesEqual (same length)In the same ratio (scale factor k )

SizeSameMay differ

Scale factork = 1any k > 0

Criteria (triangles)SSS SAS ASA RHSAAA all sides in ratio SAS-similar

RelationshipCongruent is a special case of similar with k = 1.

Language trap

Students sometimes call similar shapes "congruent" because they have the same angles. Remember: congruent requires same size. If the sides aren't equal, they're similar, not congruent.

Higher Preview: Area & Volume SF

HIGHER TIER EXTENSION · G19

G19 extends at Higher to area and volume relationships in similar figures. This is covered in full in Lesson 4.08. Here's the rule that ties it all together:

length SF = k → area SF = k² → volume SF = k³

Quick intuition

Length is one-dimensional — sides scale by k.

Area is two-dimensional — it scales by k × k = k².

Volume is three-dimensional — it scales by k × k × k = k³.

One-line example

Double the length (k = 2) → quadruple the area (k² = 4) → eight times the volume (k³ = 8).

⚠️ HIGHER EXAMINER ALERT. The biggest trap on area/volume similarity questions is applying the linear SF directly to an area or volume. Always square for area, cube for volume. Full practice in Lesson 4.08.

Summary

Key Takeaways

Congruent — same shape, same size. Corresponding sides AND angles equal.

Similar — same shape, different size. Corresponding angles equal; sides in same ratio.

Scale factor k = new length ÷ original length. Apply to every corresponding pair.

Congruent is just similar with k = 1.

Congruence criteria

SSSSASASARHS. Always state the code — never "they look the same".

Similarity criteria

AAA (usually via shared/parallel-line angles), all sides in ratio, or SAS-similar.

Top exam tip

Always name corresponding pairs first — then compute the scale factor, then apply it. Pairs by angle (arc marks) or by vertex-label order. Getting the pairing wrong is the single biggest source of lost marks in this topic.