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# Congruent Triangles
- URL: https://www.esheets.io/congruent-triangles/
- Published: 2026-10-24T19:31:04.000Z
- Updated: 2026-10-24T19:51:29.000Z
- Author: Richard Linnington
- Tags: Maths, Geometry and Shape

Practise deciding when two triangles must be congruent using SSS, SAS, ASA and RHS. You will also reject insufficient information and complete short geometric proofs involving parallelograms, angle facts and tangents to a circle. [Jump to the questions](#practise-now)

## Practise now

Worksheet preview and key skills 

### Worksheet preview

Eight self-marking questions progress from recognising congruence criteria to completing short proofs.

Each new set selects from carefully checked diagrams, including rotated triangles, a parallelogram and tangents to a circle.

#### What you’ll practise

- Recognising SSS, SAS, ASA and RHS.
- Explaining why AAA or SSA does not guarantee congruence.
- Completing concise proofs with shared sides, angle sums and tangent facts.

Use the interactive worksheet below, or read the [Topic guide](#topic-guide) for the method and worked example.

Enter the requested criterion or proof step. Each question is worth one mark and must be fully correct.

## Topic guide

Congruent triangles have exactly the same size and shape. Their matching sides have equal lengths and their matching angles are equal, even when one triangle has been rotated or reflected.

### The four criteria

- **SSS:** all three pairs of corresponding sides are equal.
- **SAS:** two pairs of corresponding sides and the angle included between them are equal.
- **ASA:** two pairs of corresponding angles and the included side are equal. A missing angle may first be found using the angle sum of a triangle.
- **RHS:** both triangles are right-angled, their hypotenuses are equal and one other pair of corresponding sides is equal.

AAA fixes shape but not size, so it proves similarity rather than congruence. SSA is also insufficient in general: two sides and a non-included angle can sometimes make different triangles.

### Worked parallelogram proof

In parallelogram *ABCD*, diagonal *AC* forms triangles *ABC* and *CDA*. Opposite sides give *AB = CD* and *BC = DA*. The diagonal is common, so *AC = CA*. Therefore triangle *ABC* is congruent to triangle *CDA* by SSS. Notice that the vertex order records the correspondence A↔C, B↔D and C↔A.

### RHS with tangents

Suppose tangents from an external point *B* touch a circle at *A* and *C*, and *O* is the centre. A radius is perpendicular to a tangent at the point of contact, so angles *OAB* and *OCB* are right angles. Also *OA = OC* because they are radii, and *OB* is the shared hypotenuse. Thus triangles *OAB* and *OCB* are congruent by RHS.

### Useful checks

- Match vertices before writing triangle names.
- For SAS, check that the stated angle lies between the two stated sides.
- For RHS, identify the side opposite each right angle: that is the hypotenuse.
- Do not assume equal lengths or angles merely because a diagram looks symmetrical.