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Congruent Triangles

FoundationHigherAQAEdexcelOCR

Congruent Triangles is a key geometry topic at GCSE Maths. This Foundation and Higher worksheet gives you exam-style questions on proving triangles are congruent, with a full mark scheme so you can check every method mark. Suitable for AQA, Edexcel and OCR. Download the free PDF and answers below. Prove congruence using SSS, SAS, ASA or RHS.

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These worksheets and mark schemes are original, written for Virtus Academy and checked against the current AQA, Edexcel and OCR specifications. Every worksheet comes with a full mark scheme.

Foundation

WorksheetFoundation · PDFMark SchemeFoundation · PDF

Higher

WorksheetHigher · PDFMark SchemeHigher · PDF

Topic overview

Two triangles are congruent when they are identical in shape and size, so one could be placed exactly on top of the other.

Four conditions prove congruence: SSS, where all three sides match; SAS, where two sides and the included angle match; ASA, where two angles and a corresponding side match; and RHS, where two right-angled triangles share the hypotenuse and one other side.

The condition SSA is not sufficient, because two different triangles can share two sides and a non-included angle. Naming the correct condition is required for the mark, so learn the four abbreviations precisely.

Revision notes

The four conditions

SSS: three sides. SAS: two sides and the angle between them. ASA: two angles and a corresponding side. RHS: right angle, hypotenuse and one side.

Each must be quoted by name in an exam answer.

Why SSA fails

Two sides and a non-included angle can produce two different triangles, so it does not prove congruence.

The angle in SAS must be the one between the two sides.

Writing a congruence proof

State each matching pair with a reason, then name the condition.

For example: \(AB = DE\) given, \(BC = EF\) given, angle \(B\) = angle \(E\) given, therefore the triangles are congruent by SAS.

Key points

  • Congruent triangles are identical in shape and size.
  • SSS: all three sides match.
  • SAS: two sides and the included angle.
  • ASA: two angles and a corresponding side.
  • RHS: right angle, hypotenuse and one side.
  • SSA does not prove congruence.

Worked examples

Example 1

Two triangles have all three sides equal. Which condition proves congruence?

Working

\[\text{All three sides match}\]identify what is given
\[\text{SSS}\]name the condition

Example 2

Two right-angled triangles share the same hypotenuse and one other side. Which condition applies?

Working

\[\text{Right angle, hypotenuse and a side}\]identify what is given
\[\text{RHS}\]name the condition

Example 3

Explain why SSA does not prove congruence.

Working

\[\text{The angle is not between the two sides}\]identify the problem
\[\text{Two different triangles are possible}\]so congruence is not guaranteed

Common mistakes

  • Using SSA as a condition.

    It is not sufficient, because two different triangles can satisfy it.

  • Using the wrong angle in SAS.

    The angle must lie between the two sides given.

  • Not naming the condition.

    The mark requires the abbreviation, such as SAS or RHS.

  • Confusing congruent with similar.

    Congruent means identical in size; similar allows different sizes.

Exam tips

  • Learn the four conditions and their abbreviations precisely.
  • Check the angle is included before using SAS.
  • List each matching pair with its reason before naming the condition.
  • Never use SSA.

Key terms

Congruent
Identical in shape and size.
Included angle
The angle between two given sides.
Hypotenuse
The longest side of a right-angled triangle.
Corresponding
Matching between the two triangles.

Related topics

Written and reviewed against the current AQA, Edexcel and OCR specifications. Spotted an error? Let us know.