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19.4 Congruence 1

GCSE Maths Foundation · Edexcel · Chapter 19: Congruence, Similarity and Vectors

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: congruent shapes and the four conditions for congruent triangles. Watch the video, read the explanation and worked examples, then try the revision questions. You can work through the full lesson, with interactive activities, practice quizzes and printable worksheets, in our free online course.

What you will learn

  • Recognise congruent shapes, including ones that are rotated or reflected
  • Use matching sides and angles to find missing values in congruent shapes
  • Know the conditions SSS, SAS, ASA and RHS for congruent triangles
  • Decide which condition, if any, shows that two triangles are congruent

Key words

  • Congruent: Exactly the same shape and size. One shape fits exactly on top of the other (it may need turning over).
  • Corresponding: In the same position in both shapes. Corresponding sides and angles are equal in congruent shapes.
  • Included angle: The angle between two given sides.
  • Hypotenuse: The longest side of a right-angled triangle, opposite the right angle.

What does congruent mean?

Shapes are congruent if they are identical in shape and size. A shape that has been translated, rotated or reflected is still congruent to the original. An enlargement is similar, but not congruent (unless the scale factor is 1).

Worked example: using corresponding parts

AB matches PQ and angle A matches angle P. BC matches QR.
Congruent shapes have equal corresponding sides, so x = QR = BC = 4.4 cm.

Answer: x = 4.4 cm

The four conditions

You do not need to check every side and angle. Two triangles are congruent if they match in any one of these ways:

  • SSS; Means side, side, side; What must match all three sides
  • SAS; Means side, angle, side; What must match two sides and the angle between them
  • ASA; Means angle, side, angle; What must match two angles and a corresponding side (AAS also works, because the third angle is then known)
  • RHS; Means right angle, hypotenuse, side; What must match a right angle, the hypotenuse and one other side

Worked example: which condition?

Triangle 1: sides 5 cm and 7 cm with an angle of 40° between them.
Triangle 2: sides 7 cm and 5 cm with an angle of 40° between them.
Two sides and the included angle match.

Answer: Congruent (SAS)

When the information is not enough

Two cases do not prove congruence:

  • AAA: three equal angles only show the triangles are similar. One could be an enlargement of the other.
  • SSA: two sides and an angle that is not between them. Two different triangles can sometimes be drawn.

Worked example: is it enough?

Triangle X has sides 6 cm and 9 cm, and a 30° angle opposite the 6 cm side.
Triangle Y has the same measurements.
The 30° angle is not between the two sides, so this is SSA.

Answer: Not enough to be sure they are congruent

Using congruence to find missing values

Once you know two triangles are congruent, every pair of corresponding sides and angles is equal. Match the parts carefully: use the marks or the order of the letters.

Worked example: missing angle

Triangles DEF and KLM are congruent, with D matching K, E matching L and F matching M.
Angle D = 72° and angle E = 48°. Find angle M.
Angle M = angle F = 180° − 72° − 48° = 60°

Answer: 60°

Common mistakes

  • Thinking AAA proves congruence. It only proves the triangles are similar.
  • Using SAS when the angle is not between the two sides.
  • Matching sides by their position on the page instead of by the marks or letters. A rotated or reflected triangle is still congruent.
  • Writing just "they look the same". You must give the condition and say which parts are equal.

Revision questions

Try these, then check your answers below each one.

Question 1. Two triangles have all three pairs of sides equal. Which condition shows they are congruent? (Choose from: SSS, SAS, ASA, RHS.)

Answer: SSS

Working: Three pairs of equal sides: SSS.

Question 2. Two right-angled triangles each have a hypotenuse of 13 cm and another side of 5 cm. Which condition shows they are congruent? (Choose from: SSS, SAS, ASA, RHS.)

Answer: RHS

Working: Right angle, hypotenuse and one side: RHS.

Question 3. True or false? Two triangles that both have angles 40°, 60° and 80° must be congruent.

Answer: False

Working: Equal angles only show the triangles are similar. One could be bigger, so they need not be congruent.

Related lessons

Continue in the full course

This lesson is section 19.4 (EGCMF1904) of our free GCSE Maths Foundation course. In the full course you get every video for this section, an animated interactive lesson, a self-marking practice quiz, a printable worksheet with a mark scheme, and your progress saved. Go to the full GCSE Maths Foundation course.

 

 

 

 

 

 

 

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