Geometry & Measures
Grade 5-7Congruence
Two shapes are congruent if they are exactly the same shape and size, and GCSE questions test whether you can identify which condition proves two triangles are congruent, and use congruent triangles to build a formal geometric proof. This lesson covers the four congruence conditions, writing a full congruence proof, and using congruence to find missing lengths and angles.
Written by Asad, Co-Founder of Teachably
Video walkthrough coming soon
The written lesson below covers everything you need in the meantime.
The four congruence conditions
Two triangles are congruent if one of four conditions holds: SSS (all three sides equal), SAS (two sides and the angle between them equal), ASA (two angles and the side between them equal), or RHS (a right angle, the hypotenuse, and one other side equal, for right-angled triangles only). Any one of these being true guarantees the triangles are identical in every other measurement too.
If a right-angled triangle has hypotenuse 5 cm and one other side 4 cm, and a second right-angled triangle has the same measurements, the two triangles are congruent by RHS.
Why SSA does not work
Giving two sides and a non-included angle (SSA) does not guarantee congruence, since the third side can sometimes swing to two different positions while still matching the given measurements, producing two genuinely different triangles.
Triangle 1 has AB = 8 cm, BC = 5 cm and angle A = 30°, and triangle 2 has the same three measurements, but the triangles are not necessarily congruent, since angle A is opposite side BC rather than included between the two given sides.
Writing a congruence proof
A full congruence proof states each piece of equal information with its reason (given, common side, vertically opposite angles, alternate angles, and so on), then names the correct congruence condition and concludes which triangles are congruent.
In a kite ABCD with AB = AD and CB = CD, triangle ABC and triangle ADC share side AC, so all three pairs of sides are equal (AB = AD, CB = CD, AC = AC common), proving the triangles congruent by SSS.
Using congruence to find missing values and prove properties
Once two triangles are shown to be congruent, every pair of corresponding sides and angles must be equal, which can be used to find a missing length or angle, or to prove a property of a larger shape such as a parallelogram or rectangle.
In a parallelogram ABCD with diagonals crossing at O, triangle AOB is congruent to triangle COD (ASA, using alternate angles from the parallel sides), so AO = CO and BO = DO, proving that the diagonals of a parallelogram bisect each other.
Worked Examples
Three exam-style questions, fully solved.
The diagram shows two triangles. Two sides and the angle between them are given for each triangle, and the corresponding measurements are equal. Which condition proves the triangles are congruent?
Easy- 1.Identify what is given: two sides and the included angle
Answer: SAS (side, angle, side)
In the diagram, AB = AC, and AD bisects angle BAC, meeting BC at D. Prove that triangle ABD is congruent to triangle ACD.
Medium- 1.State the first pair of equal sides: AB = AC (given)
- 2.State the pair of equal angles: angle BAD = angle CAD (AD bisects angle BAC, given)
- 3.State the common side: AD = AD
- 4.Conclude using the correct condition: triangle ABD is congruent to triangle ACD (SAS)
Answer: Triangle ABD ≅ triangle ACD (SAS)
ABCD is a parallelogram, so AB = CD and AB is parallel to CD. The diagonals AC and BD intersect at O. Prove that triangle AOB is congruent to triangle COD, and hence that the diagonals bisect each other.
Hard- 1.State the equal sides: AB = CD (opposite sides of a parallelogram)
- 2.State the first pair of equal angles: angle OAB = angle OCD (alternate angles, since AB is parallel to CD)
- 3.State the second pair of equal angles: angle OBA = angle ODC (alternate angles, since AB is parallel to CD)
- 4.Conclude using ASA, then use the equal corresponding sides to finish the proof
Answer: Triangle AOB ≅ triangle COD (ASA), so AO = CO and BO = DO: the diagonals bisect each other
Avoid These
The most common mistakes students make.
Treating SSA (two sides and a non-included angle) as a valid congruence condition, when it does not guarantee the triangles are identical.
Forgetting to state a reason for each piece of equal information in a proof, such as "common side" or "vertically opposite angles", rather than just listing the equal measurements.
Matching up corresponding vertices incorrectly, for example assuming triangle ABC being congruent to triangle DEF means AB corresponds to DF rather than DE.
Naming the wrong congruence condition, such as calling a proof SSS when one of the three pieces of information given is actually an angle.
Stopping a proof once congruence is shown, without going on to state the conclusion that was actually asked for, such as that two lengths are equal or a line bisects an angle.
FAQ
Questions parents and students ask.
Before this topic, make sure you know
What to learn next
Want a plan built around your child specifically?
Get our free 8-video course, or book a free Roadmap Call for a personalised plan.