Congruence & similarity

What you will learn

recognise when two shapes are congruent (identical in shape and size),
recognise when two shapes are similar (identical shape but possibly different size),
apply the congruence tests for triangles: SSS, SAS, ASA, RHS,
apply similarity tests: AA (equal angles) or matching side ratios,
use congruent triangles to prove simple properties of quadrilaterals.

1. The easy version first — same shape, same size?

Before any rules, here is the idea.

Two triangles. The left pair look identical; the right pair look like a small and a big version of the same shape.

Congruent means same shape and same size. If you cut one out and placed it on the other, they would match exactly.
Similar means same shape but (possibly) different size. One is a scaled copy of the other — it has been enlarged or shrunk.

Every congruent pair is also similar (the scale factor is just $1$ ).

2. A very easy example

Worked example A Which are congruent?

A builder cuts three triangles. Triangle 1 has sides $3, 4, 5$ cm. Triangle 2 has sides $3, 4, 5$ cm. Triangle 3 has sides $6, 8, 10$ cm.

Triangles 1 and 2 have all three sides the same, so they are congruent (identical).

Triangle 3 has sides double those of Triangle 1 — all in the same ratio. So Triangle 1 and Triangle 3 are similar, with scale factor $2$ .

3. Four ways to prove two triangles are congruent

Instead of checking all six measurements (three sides and three angles), any one of these four combinations is enough.

SSS — three sides match

SSS: all three pairs of sides are equal.

SAS — two sides and the angle between them

SAS: two sides and the included angle are equal. 'Included' means the angle between the two named sides.

ASA — two angles and the side between them

ASA: two angles and the included side are equal.

RHS — right angle, hypotenuse and side

RHS: both have a right angle; the hypotenuses match; one pair of the other sides matches.

The four congruence tests at a glance

SSS — side, side, side

All three pairs of sides are equal.

SAS — side, angle, side

Two pairs of sides and the included angle (the angle between those two sides) are equal.

ASA — angle, side, angle

Two pairs of angles and the included side are equal.

RHS — right angle, hypotenuse, side

Both triangles have a right angle, the hypotenuses are equal, and one pair of the other sides is equal.

4. Applying a test

Worked example E Congruent by SSS — the simplest case

Two triangles both have sides $3$ , $4$ , $5$ cm. Are they congruent?

All three pairs of sides match ( $3 = 3$ , $4 = 4$ , $5 = 5$ ), so the triangles are congruent by SSS.

Worked example 1 Which test applies?

In triangles $ABC$ and $DEF$ : $AB = DE = 7$ cm, $AC = DF = 9$ cm, and $\angle A = \angle D = 45^\circ$ .

Work out which two sides and which angle are named, and whether the angle sits between the two sides.

In $\triangle ABC$ : sides $AB$ and $AC$ both touch vertex $A$ . The named angle is $\angle A$ — sitting between them. ✓
Same for $\triangle DEF$ .

So two pairs of sides and the included angle are equal. The test is SAS — the triangles are congruent.

Worked example 2 Why SSA can fail (watch-out)

Suppose $AB = DE = 6$ , $BC = EF = 4$ , $\angle A = \angle D = 30^\circ$ (not between the two given sides).

Two triangles fitting those numbers exist: a “narrow” one and a “wide” one, depending on how you swing side $BC$ from $B$ . So this data is not enough to prove congruence.

Lesson: the angle has to sit between the two sides for SAS to work.

5. Similar triangles

If two triangles have the same shape but possibly different sizes, they are similar. Matching angles are equal; matching sides are in the same ratio.

Similar triangles: matching angles equal; matching sides in a 1:2 ratio.

Tests for similarity

AA — two pairs of equal angles

If two pairs of corresponding angles match, the triangles are similar. (The third angle must also match, because the three angles sum to $180^\circ$ .)

SSS similarity — three sides in ratio

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

SAS similarity — two sides in ratio + included angle

Two pairs of corresponding sides are in the same ratio and the included angle is equal.

Worked example E2 Spotting similarity — doubled sides

Triangle $A$ has sides $3, 4, 5$ . Triangle $B$ has sides $6, 8, 10$ . Are they similar?

Every side of $B$ is exactly twice the matching side of $A$ : $\frac{6}{3} = \frac{8}{4} = \frac{10}{5} = 2$ . The ratio is constant, so the triangles are similar with scale factor $2$ .

Worked example 3 Scale factor

Triangles $ABC$ and $PQR$ : $AB = 6$ , $PQ = 9$ ; $AC = 10$ , $PR = 15$ ; $\angle A = \angle P$ .

Check the ratios: $\dfrac{PQ}{AB} = \dfrac{9}{6} = 1.5$ and $\dfrac{PR}{AC} = \dfrac{15}{10} = 1.5$ . Same ratio, and the included angles are equal.

So $\triangle ABC \sim \triangle PQR$ by SAS similarity with scale factor $1.5$ (the second triangle is $1.5\times$ the first).

6. Using congruent triangles (quadrilateral properties)

Many quadrilateral facts — “opposite sides of a parallelogram are equal”, “diagonals of a rhombus bisect each other” — are proved by splitting the shape into two triangles and showing those triangles are congruent.

Shape	Diagonals
Parallelogram	bisect each other
Rectangle	bisect each other; equal
Rhombus	bisect each other; perpendicular
Square	bisect each other; equal; perpendicular
Kite	perpendicular

7. A classification algorithm

Given two triangles:

All three pairs of sides equal? → Congruent (SSS).
Two pairs of sides in the same ratio and the included angles equal? → Similar (SAS), possibly congruent if scale is $1$ .
Two pairs of angles equal? → Similar (AA).
None of the above → Neither.

Practice: Year 8 core

Fluency

Congruence tests

For each, state which congruence test applies (SSS, SAS, ASA, RHS) or explain why none do.

Two triangles with sides $3, 3, 3$ and $3, 3, 3$ . (Warm-up — what test is this?)
A right-angled triangle with hypotenuse $5$ and leg $3$ , and another right-angled triangle with hypotenuse $5$ and leg $3$ . (Which test uses a right angle?)
Two triangles with sides $4, 5, 6$ and $4, 5, 6$ .
Two triangles: $\angle A = 30^\circ$ , $AB = 5$ , $BC = 6$ ; $\angle D = 30^\circ$ , $DE = 5$ , $EF = 6$ . (Try mentally first: is the $30^\circ$ angle between the two named sides?)
Two right-angled triangles: hypotenuse $10$ with one leg $6$ , vs hypotenuse $10$ with one leg $6$ .
Two triangles: $\angle A = 40^\circ$ , $AB = 5$ , $\angle B = 60^\circ$ ; $\angle D = 40^\circ$ , $DE = 5$ , $\angle E = 60^\circ$ .
Two triangles: $AB = 7$ , $BC = 5$ , $\angle C = 40^\circ$ ; $DE = 7$ , $EF = 5$ , $\angle F = 40^\circ$ .

Fluency

Similar triangles

$\triangle ABC$ has sides $3, 4, 5$ . $\triangle DEF$ has sides $9, 12, 15$ . Are they similar? What is the scale factor?
$\triangle ABC$ has $\angle A = 40^\circ$ , $\angle B = 70^\circ$ . $\triangle PQR$ has $\angle P = 40^\circ$ , $\angle R = 70^\circ$ . Similar? Why?
Two triangles have sides in ratio $2 : 3 : 4$ and $4 : 6 : 8$ . Similar?
Triangle $A$ has sides $5, 6, 8$ . Triangle $B$ has sides $10, 12, 15$ . Similar?

Reasoning

Explain and spot the mistake

Sam says two right-angled triangles with legs $3, 4$ each must be congruent. Is Sam right? Explain.
Mira writes “SSA is a valid test for congruence because three measurements are given”. Is this correct? Give a counter-example.
Explain why two similar triangles with scale factor $1$ are also congruent.
A rhombus has all four sides equal. Prove (with triangle congruence) that its diagonals bisect each other.

Problem solving

Real contexts

A ramp has a shadow $3$ m long when a $1$ -m pole casts a $0.5$ -m shadow. How high is the ramp? (Use similar triangles.)
A photo is enlarged: $10$ cm by $15$ cm becomes $30$ cm by $45$ cm. Find the scale factor and the area ratio.
Two triangles on a flag each have sides in ratio $5 : 12 : 13$ . Are they congruent or similar? What extra information do you need?
A tall tree’s shadow is $6$ m at the same moment a $1.8$ m friend’s shadow is $1.2$ m. How tall is the tree?

Challenge

Reasoning

Harder reasoning

Explain why two isosceles triangles with equal apex angles and one pair of equal sides may still not be congruent.
In a parallelogram $ABCD$ , show using congruent triangles that $AB = CD$ and $AD = BC$ .
A triangle has sides $6, 8, 10$ . A similar triangle has a hypotenuse $15$ . Find its other two sides.
Are all squares similar? Are all rectangles similar? Justify.