Congruent Triangles
This topic covers the four key criteria for proving that two triangles are identical in shape and size (congruent). Understanding these rules allows you to deduce unknown lengths and angles in geometric problems without direct measurement.
Part of the ESAT Mathematics 1 syllabus — revision for the Engineering and Science Admissions Test (ESAT), the UAT-UK admissions test for Cambridge, Imperial, Oxford and UCL.
Key points
- Congruent triangles are exact duplicates; all corresponding sides and angles are equal.
- SSS (Side-Side-Side): If all three sides of one triangle are equal in length to the corresponding three sides of another, they are congruent.
- SAS (Side-Angle-Side): If two sides and the angle *between* them in one triangle match those in another, they are congruent.
- ASA (Angle-Side-Angle): If two angles and a corresponding side in one triangle match those in another, they are congruent. The side can be between the angles or another corresponding side.
- RHS (Right-angle-Hypotenuse-Side): If two right-angled triangles have hypotenuses of the same length and one other pair of corresponding sides are equal, they are congruent.
- Note that SSA (Side-Side-Angle) is NOT a valid congruence criterion, as it can lead to two different possible triangles (ambiguous case), unless the angle is 90 degrees (which is the RHS case).
Diagram
Formulae
SSS (Side-Side-Side) Use when you know the lengths of all three sides of two triangles.
SAS (Side-Angle-Side) Use when you know two side lengths and the angle created between them.
ASA (Angle-Side-Angle) Use when you know two angles and the length of one corresponding side.
RHS (Right-angle-Hypotenuse-Side) Use ONLY for right-angled triangles when you know the hypotenuse and one other side.
Definitions
- Congruent
- Two geometric figures are congruent if they have the same shape and size. For triangles, this means all three corresponding sides and all three corresponding angles are equal.
- Hypotenuse
- The longest side of a right-angled triangle, which is always the side opposite the right angle.
- Included Angle
- The angle formed between two specified sides of a triangle.
Worked example
In the diagram, a circle has centre O. Points A and B are on the circumference. M is the midpoint of the chord AB. Prove that triangle OMA is congruent to triangle OMB.
- 1
Identify known facts from the diagram and prompt.
OA and OB are both radii of the circle, so OA = OB.
- 2
The prompt states M is the midpoint of AB, which means AM = MB.
- 3
The side OM is common to both triangle OMA and triangle OMB.
- 4
We now have three pairs of equal corresponding sides:
OA = OB (radii), AM = MB (midpoint), and OM = OM (common) - 5
The condition SSS (Side-Side-Side) is met.
- 6
Therefore, triangle OMA is congruent to triangle OMB (SSS).
Answer: Triangle OMA is congruent to triangle OMB because they satisfy the SSS criterion: OA=OB (radii), AM=MB (M is the midpoint), and OM is a common side.
Common mistakes
- ×Mistaking SSA for a valid criterion. Side-Side-Angle does not guarantee congruence, except for the special RHS case.
- ×For SAS, failing to check that the known angle is the one *included* between the two known sides.
- ×For ASA, incorrectly matching the side. The equal side must be in the same relative position to the angles in both triangles (e.g., both sides are opposite the 50° angle).
- ×Assuming triangles are congruent based on their appearance in a diagram, which may not be drawn to scale. You must use only the information given.
No-calculator tips
- ✓Always draw a quick sketch of the problem and label all given information. This helps to visualise which criterion might apply.
- ✓Look for 'hidden' equal parts. For example, a shared side is common to two triangles, or vertically opposite angles are always equal.
- ✓Systematically work through the four criteria (SSS, SAS, ASA, RHS) to see which one matches the information you have and can deduce.
- ✓Once you prove congruence, remember you can deduce that all other corresponding parts are equal. For example, proving triangles are congruent via SSS also proves their corresponding angles are equal.