SSA does not guarantee congruence

The idea

When two sides and a non-included angle of two triangles are equal (the SSA condition), the triangles need not be congruent, since two non-congruent triangles can have these measurements.

Is this enough?

Rabia draws a triangle ABC with AB = 6 cm, AC = 4 cm and ∠B = 30°. Meera draws one with the same measures. Rabia says: “Two sides and an angle, as in SAS, so ours must be congruent.” Look at where the angle sits.

Must their triangles be congruent?

• Yes, two sides and an angle are equal
• No, they could be different triangles
• Only if both are drawn the same way up

Find the corner C

AB = 6 cm is drawn and the 30° angle at B fixes a ray. The corner C must be on the ray and 4 cm from A. Slide C along the ray. How many places work?

Two different triangles

Both triangles have AB = 6 cm, AC = 4 cm and ∠B = 30°. But their third side BC is different, so one cannot fit on the other.

If two sides and an angle that is not between them of one triangle are equal to those of another (the SSA condition), the triangles need not be congruent: two different triangles can have the same measures.

• Triangle | AB | AC | ∠B | BC
• ABC | 6 cm | 4 cm | 30° | 2.6 cm
• ABD | 6 cm | 4 cm | 30° | 7.8 cm

Notes

If two sides and an angle that is not between them of one triangle are equal to those of another (the SSA condition), the triangles need not be congruent: two different triangles can have the same measures.

Check yourself

∆ABC and ∆XYZ have AB = XY = 6 cm, AC = XZ = 4 cm and ∠B = ∠Y = 30°. Are they congruent?

How many different, non-congruent triangles have AB = 6 cm, AC = 4 cm and ∠B = 30°?

Answer: 2

The corner C can be at 2 places, giving triangles with BC = 2.6 cm and BC = 7.8 cm. So there are 2 different triangles.

Now AB = 6 cm, ∠B = 30° and AC = 2 cm. The nearest point of the ray to A is 3 cm away. How many triangles can we draw?

Meera says: “SAS and SSA both use two sides and an angle, so both prove congruence.” What is the mistake?

• Not necessarily. They may be different triangles. — correct. Yes! This is SSA. The arc of 4 cm cuts the ray twice, so two different triangles fit.
• Yes, by SAS. SAS needs the angle between the two sides. ∠B is not between AB and AC.
• Yes, two sides and one angle is always enough. It is not. We found two triangles with the same two sides and the same angle, but different BC.
• None. The arc is too short to reach the ray. — correct. Yes! A is at least 3 cm from every point of the ray. A corner 2 cm from A does not exist on it.
• 2 triangles. A 2 cm arc from A is too short to reach the ray, as the nearest point is 3 cm away.
• 1 triangle. The 2 cm arc does not touch the ray at all. It would need to be at least 3 cm.
• In SAS the angle is between the sides. In SSA it is not, and SSA does not guarantee congruence. — correct. Yes! The position of the angle matters. Between the two sides it fixes the triangle. Not between them, it may not.
• SSA is the same as SAS. They are different. In SAS the angle is between the two sides; in SSA it is not.
• There is no mistake. There is. SSA can give two different triangles, as we saw.