Year 8 - Congruence and similarity - Free practice

Year 8 Congruence & Similarity Worksheets

SSS, SAS, AAS - which conditions prove triangles match?

Quick answer

What are the conditions for congruence and similarity of triangles?

Two triangles are congruent if they match by SSS (three equal sides), SAS (two sides and the included angle), AAS (two angles and a matching side), or RHS (right angle, hypotenuse, one side). They are similar if they match by AA (two equal angles), SSS in ratio, or SAS in ratio. Similar triangles have proportional sides, so you can use a scale factor to find missing lengths.

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Worked examples

Year 8 congruence and similarity: worked examples

Three examples, from a standard question to a worded problem. Have a go at each one before you open the solution.

Example 1 · Identifying a congruence condition (SAS)
Triangle $ABC$ has $AB = 5$ cm, $BC = 8$ cm and $\angle ABC = 47°$ . Triangle $PQR$ has $PQ = 5$ cm, $QR = 8$ cm and $\angle PQR = 47°$ . State whether the two triangles are congruent and give the condition that applies.
Show worked solution
Key idea: Two triangles are congruent if they satisfy one of the four conditions: SSS, SAS, AAS or RHS.
1. List the matching information: $AB = PQ = 5$ cm, $BC = QR = 8$ cm, and the included angles $\angle ABC = \angle PQR = 47°$ .
2. The equal angle is between the two pairs of equal sides in each triangle.
3. This matches the SAS (Side-Angle-Side) condition.
4. Therefore $\triangle ABC \equiv \triangle PQR$ (SAS).
Answer: The triangles are congruent by SAS.
Example 2 · Using similarity to find a missing side length
Triangle $DEF$ is similar to triangle $XYZ$ , with $\angle D = \angle X$ and $\angle E = \angle Y$ . The side lengths of $\triangle DEF$ are $DE = 6$ cm, $EF = 9$ cm and $DF = 12$ cm. In $\triangle XYZ$ , $XY = 10$ cm. Find the length of $YZ$ .
Show worked solution
Key idea: In similar triangles, matching sides are in the same ratio (the scale factor).
1. Identify the matching sides: $DE$ matches $XY$ , and $EF$ matches $YZ$ .
2. Find the scale factor: $k = \frac{XY}{DE} = \frac{10}{6} = \frac{5}{3}$
3. Apply the scale factor to find $YZ$ : $YZ = EF \times k = 9 \times \frac{5}{3} = 15 \text{ cm}$
4. Check: The ratio $\dfrac{YZ}{EF} = \dfrac{15}{9} = \dfrac{5}{3}$ matches the scale factor. ✓
Answer: $YZ = 15$ cm
Example 3 · Real-life problem: using similar triangles to find a height
Layla stands next to a flagpole on flat ground. She is 1.6 m tall and casts a shadow 2.0 m long. At the same time, the flagpole casts a shadow 15.0 m long. Layla's triangle (formed by her height and shadow) and the flagpole's triangle (formed by its height and shadow) are similar because they share the same sun angle. Find the height of the flagpole.
Show worked solution
Key idea: In similar triangles, matching sides are in the same ratio.
1. The matching sides are: Layla's height matches the flagpole's height, and Layla's shadow matches the flagpole's shadow.
2. Write the ratio of matching sides: $\frac{\text{flagpole height}}{\text{Layla's height}} = \frac{\text{flagpole shadow}}{\text{Layla's shadow}}$
3. Substitute the known values: $\frac{h}{1.6} = \frac{15.0}{2.0}$
4. Solve for $h$ : $h = 1.6 \times \frac{15.0}{2.0} = 1.6 \times 7.5 = 12 \text{ m}$
5. Check: $\dfrac{12}{1.6} = 7.5$ and $\dfrac{15.0}{2.0} = 7.5$ ✓
Answer: The flagpole is 12 m tall.

Common mistakes

Where students usually lose marks

Mixing up included and non-included angles

Claiming SSA (two sides and a non-included angle) proves congruence.

SSA is not a valid congruence condition because two different triangles can share the same two side lengths and the same non-included angle. The angle in SAS must sit directly between the two given sides.

Matching sides in the wrong order for similar triangles

Writing $\dfrac{DE}{XY} = \dfrac{EF}{XZ}$ when $\angle E = \angle Y$ (so $EF$ should match $YZ$ , not $XZ$ ).

Always pair sides that are opposite (or adjacent to) equal angles. Write the vertices of both triangles in matching order, then read off the ratios from those matching positions.

Confusing the congruence symbol with the similarity symbol

Writing $\triangle ABC \sim \triangle PQR$ when the triangles are the same size and shape (congruent).

Use $\equiv$ for congruence (same shape and same size) and $\sim$ for similarity (same shape, possibly different size). Congruent triangles are a special case of similar triangles with a scale factor of 1.

Forgetting to state the condition in a congruence proof

Concluding "therefore $\triangle ABC \equiv \triangle PQR$ " without writing which condition (SSS, SAS, AAS or RHS) was used.

A formal congruence proof always ends with the condition in brackets, for example: $\triangle ABC \equiv \triangle PQR$ (AAS). Without the reason, the proof is incomplete.

Try one yourself

Your turn

Triangle $ABC$ is similar to triangle $DEF$ . The sides of triangle $ABC$ are $AB = 4$ cm, $BC = 6$ cm, and $AC = 8$ cm. The side $DE = 6$ cm corresponds to $AB = 4$ cm. Find the length of $EF$ in centimetres, which corresponds to $BC$ .

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About this worksheet

Why we built it

Year 8 introduces the four congruence conditions (SSS, SAS, AAS, RHS) and the corresponding similarity conditions. The hard part is recognising which one applies - these worksheets give your student varied figures to train that instinct.

What's covered

Sub-skills your student will practise

✓The four triangle-congruence conditions (SSS, SAS, AAS, RHS)
✓Identifying which condition applies to a given figure
✓Similar-triangle conditions (AA, SSS, SAS in ratio)
✓Using similarity to find missing side lengths
✓Short formal proofs of congruence

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Frequently asked

Questions parents ask about congruence and similarity

What's the difference between congruent and similar?

Congruent shapes are exactly the same - same side lengths and angles. Similar shapes have the same angles but their sides are in proportion - one is a scaled version of the other. Photos enlarged on a printer are similar, not congruent.

What are the four triangle congruence conditions?

SSS (three equal sides), SAS (two sides and the angle between them), AAS (two angles and a non-included side), and RHS (right angle, hypotenuse, side). Any one of these is enough to prove congruence.

Why doesn't SSA prove congruence?

Because two triangles can share two sides and a non-included angle yet have different shapes. This is the 'ambiguous case'. RHS works because the right angle removes the ambiguity.

How is similarity used in Year 8?

Mostly to find missing lengths. If two triangles are similar with a ratio of 2:3, every pair of corresponding sides has the same ratio. This sets up trigonometry in Year 9.

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