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GCSE Maths revision · Geometry and measures

Congruent triangles

Foundation and Higher

The idea

Two triangles are congruent when they are exactly the same shape and size. Prove it with one of four conditions: SSS, SAS, ASA (or AAS) and RHS.

  1. Find three pairs of matching facts, with a reason for each.
  2. Check they fit a condition, for SAS the angle must be between the two sides.
  3. State the condition at the end: congruent (SAS).

Watch out Three equal angles (AAA) show triangles are similar, not congruent.

See it worked

Worked exampleNon-calculator1 mark

Each of two triangles has sides of length 7 cm and 11 cm. The angle between these two sides is 38∘38^\circ in each triangle. Which congruence test proves that the triangles are congruent?

  1. The 38∘38^\circ angle lies between the given 7 cm and 11 cm sides.
  2. Two sides and the included angle establish SAS congruence.

Answer SAS

Your turn

Question 1Non-calculator3 marks

Each part describes two triangles. Choose the condition that shows they are congruent.

(a) Triangle 1 has sides 5 cm and 7 cm with an angle of 40∘40^\circ between them. Triangle 2 also has sides 5 cm and 7 cm with an angle of 40∘40^\circ between them.

(b) Both triangles have a right angle, a hypotenuse of 13 cm and another side of 5 cm.

(c) Explain why two triangles with the same three angles need not be congruent.

Write your answer, then show the working
  1. (a) Two sides and the angle between them (the included angle) match: SAS.
  2. (b) Right angle, hypotenuse and one other side: RHS.
  3. (c) An enlargement keeps every angle but changes the side lengths.

Answer (a) SAS; (b) RHS; (c) They can be different sizes: equal angles only make them similar.

Question 2Non-calculator4 marks

Two triangles each have angles of 40∘40^\circ and 75∘75^\circ. Each triangle has one side of length 9 cm.

(a) Find the third angle of each triangle.

(b) Explain why the triangles need not be congruent.

(c) In both triangles the 9 cm side is opposite the 40° angle. Are the triangles now congruent? Give the condition.

Write your answer, then show the working
  1. (a) Subtract the two given angles from 180°. 180−40−75180-40-75
  2. Therefore 6565°.
  3. (b) The 9 cm sides may be opposite different angles, so the equal sides might not correspond.
  4. (c) Yes, AAS: two angles and a corresponding side are equal.

Answer (a) 6565°; (b) The 9 cm sides may be opposite different angles, so the equal sides might not correspond.; (c) Yes, AAS: two angles and a corresponding side are equal.

Question 3Non-calculator3 marks

Triangle PQRPQR has PQ=7PQ = 7 cm, angle P=50∘P = 50^\circ and angle Q=60∘Q = 60^\circ. Triangle XYZXYZ has XY=7XY = 7 cm, angle X=50∘X = 50^\circ and angle Z=70∘Z = 70^\circ.

Are the two triangles congruent? You must show how you get your answer.

Choose one

In past papers

45 official questions in 44 papers since 2017. The latest:

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