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19.5 Congruence 2

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

A free GCSE Maths Foundation lesson for Edexcel (9–1) from OBA School: proving triangles are congruent and using congruence in shapes. 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

  • Write a short proof that two triangles are congruent, giving a reason for every fact
  • Use angle facts (vertically opposite, alternate) as part of a congruence proof
  • Use congruent triangles inside parallelograms, kites and isosceles triangles
  • Use congruence to show that two lengths or angles are equal

Key words

  • Proof: A chain of facts, each with a reason, that leads to a conclusion.
  • Common side: A side that belongs to both triangles. It is equal to itself, so it counts as a matching pair.
  • Alternate angles: The Z-angles made by a line crossing two parallel lines. They are equal.
  • Vertically opposite: Opposite angles where two straight lines cross. They are equal.

How to write a congruence proof

A congruence proof has three matching facts, each with a reason, and then the condition.

  • 1; What to write Name the two triangles, with the letters in matching order.
  • 2; What to write Write three pairs of equal sides or angles. Give a reason for each one ("given", "common side", "alternate angles", …).
  • 3; What to write Finish with "so the triangles are congruent (SAS)" or whichever condition you used.

Worked example: prove that triangles AMB and CMD are congruent

AM = CM (M is the midpoint of AC)
BM = DM (M is the midpoint of BD)
Angle AMB = angle CMD (vertically opposite angles)
Two sides and the included angle are equal.

Answer: Triangles AMB and CMD are congruent (SAS)

Congruence inside a parallelogram

A diagonal cuts a parallelogram into two congruent triangles. The parallel sides give pairs of alternate angles, and the diagonal is a common side.

Worked example: prove and use

Angle ABD = angle CDB (alternate angles, AB parallel to DC)
Angle ADB = angle CBD (alternate angles, AD parallel to BC)
BD is common to both triangles. So triangles ABD and CDB are congruent (ASA).
Angle ABD = 35°, so angle CDB = 35°. Angle ADB = 70°, so angle CBD = 70°.

Answer: Angle CDB = 35°, angle CBD = 70°

Right-angled triangles (RHS)

If two right-angled triangles share a side and have equal hypotenuses, they are congruent by RHS. This is useful in real-life symmetry problems.

Worked example: show that the wires reach the ground the same distance from the mast

Angle TFP = angle TFQ = 90° (the mast is vertical)
TP = TQ (the wires are equal) and TF is common.
So triangles TFP and TFQ are congruent (RHS), and PF = QF.
Using Pythagoras: PF = √(13² − 12²) = √25 = 5 m

Answer: PF = QF = 5 m

Congruent, similar or neither?

Congruent shapes are the same size. Similar shapes can be different sizes but have equal angles. Check the sides first.

  • 3 cm, 4 cm, 5 cm; Triangle 2 4 cm, 5 cm, 3 cm; Decision Congruent (SSS)
  • 3 cm, 4 cm, 5 cm; Triangle 2 6 cm, 8 cm, 10 cm; Decision Similar (× 2)
  • 3 cm, 4 cm, 5 cm; Triangle 2 3 cm, 4 cm, 6 cm; Decision Neither

Worked example: decide

Triangle P has angles 50°, 60° and a 7 cm side between them. Triangle Q has angles 60°, 50° and a 7 cm side between them.
Two angles and the side between them match.

Answer: Congruent (ASA)

Common mistakes

  • Giving facts without reasons. Every equal pair needs a reason.
  • Using "AAA" as a condition. It only shows similarity.
  • Forgetting that a shared (common) side counts as one of the three facts.
  • Mixing up alternate and corresponding angles when giving reasons.

Revision questions

Try these, then check your answers below each one.

Question 1. ABCD is a parallelogram with diagonal AC. Angle BAC = 42°. Which other angle must also be 42°? (Choose from: Angle DCA, Angle BCA, Angle ADC, Angle ABC.)

Answer: Angle DCA

Working: Angles BAC and DCA are alternate angles (AB parallel to DC), so angle DCA = 42°.

Question 2. AC and BD cross at M, and M is the midpoint of both lines. Which condition proves that triangles AMB and CMD are congruent? (Choose from: SAS, SSS, ASA, RHS.)

Answer: SAS

Working: AM = CM and BM = DM (midpoint), and angle AMB = angle CMD (vertically opposite). Two sides and the included angle: SAS.

Question 3. True or false? In isosceles triangle ABC with AB = AC, the line from A to the midpoint M of BC splits the triangle into two congruent triangles.

Answer: True

Working: AB = AC, BM = CM and AM is common, so the triangles are congruent (SSS).

Related lessons

Continue in the full course

This lesson is section 19.5 (EGCMF1905) 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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