Two triangles are congruent when matching sides and angles are all equal, and you can prove it with as few as three matching facts. The facts must be in the right positions, not just present.
This lesson opens the section on congruency, enlargement and combined transformations. The next lesson moves from sameness to size change, in calculating enlargement scale factors.
Which conditions prove congruence?
Match each condition to what you are given.
| Condition | What must be equal |
|---|---|
| SSS | All three sides |
| SAS | Two sides and the angle between them |
| ASA | Two angles and the side between them |
| AAS | Two angles and a side that is not between them |
Your textbook may list the conditions slightly differently, so compare it with this table.
Worked example: SAS
In triangle ABC, AB = 6 cm, BC = 8 cm and angle B = 40°. In triangle PQR, PQ = 6 cm, QR = 8 cm and angle Q = 40°.
Angle B sits between AB and BC, and angle Q sits between PQ and QR. Two sides and the included angle are equal, so the triangles are congruent by SAS.
The matching is written in order: A matches P, B matches Q and C matches R.
The mistake of using a non-included angle
A common slip is to call two triangles congruent when two sides and an angle are equal, but the angle is not between those sides. This is sometimes called SSA, and it does not prove congruence.
Here is an example. Let angle A = 30°, AC = 8 cm and BC = 5 cm. Then sin B = 8 × sin 30° ÷ 5 = 0.8, so angle B is about 53° or about 127°. Both give a valid triangle, so the same facts fit two different triangles.
The check is to find where the angle sits. If it is not between the two given sides, SAS does not apply.
Check yourself
In triangles DEF and XYZ, angle D = 50°, angle E = 60° and DE = 9 cm. Also angle X = 50°, angle Y = 60° and XY = 9 cm. Are they congruent? Would they be if only the three angles were known?
Answer
DE lies between angles D and E, and XY lies between angles X and Y. Two angles and the side between them are equal, so the triangles are congruent by ASA.
If only the three angles were known, the triangles would be similar but not necessarily congruent, because the size is not fixed.
What to study next
Move on to size change in calculating enlargement scale factors. Test mixed questions in the section practice set.
If you want a teacher to train the wording of your congruence reasons, see online one-to-one Mathematics tuition.