AAS condition

The idea

When two angles and a non-included side of two triangles are equal (the AAS condition), the triangles are congruent, because the angle sum of 180° makes the third angles equal too, giving ASA.

Two angles, one side

Meera has two triangles ABC and XYZ. In both, one angle is 35°, another is 75°, and the side BC = YZ = 4 cm. But the 4 cm side is not between the two known angles. Are they congruent?

Does the side that is not between the angles still fix the triangle?

• No, ASA needs the side between the angles
• Yes, the triangles are congruent
• Only if they are turned the same way

Tear the corners off

The angles of a triangle add up to 180°, the angles on a straight line too. Tear off the two known corners of each triangle and lay them on a straight line. What is left is the third angle.

The third angle is fixed too

In both triangles the gap that is left is 70°. So ∠B = ∠Y = 70°. Now BC lies between ∠B and ∠C, and YZ between ∠Y and ∠Z. That is the ASA condition, so the triangles are congruent.

If two angles and a side that is not between them of one triangle are equal to two angles and the corresponding side of another triangle, the triangles are congruent. This is the AAS (Angle Angle Side) condition. The third angles are equal too, so it becomes ASA.

• ∆ABC | ∆XYZ
• first angle | 35° | 35°
• second angle | 75° | 75°
• third angle (180° − 35° − 75°) | 70° | 70°
• side BC = YZ | 4 cm | 4 cm

Notes

If two angles and a side that is not between them of one triangle are equal to two angles and the corresponding side of another triangle, the triangles are congruent. This is the AAS (Angle Angle Side) condition. The third angles are equal too, so it becomes ASA.

Check yourself

In ∆ABC, ∠A = 40° and ∠C = 65°. What is ∠B, in degrees?

Answer: 75

∠B = 180° − 40° − 65° = 75°. If ∆XYZ has ∠X = 40° and ∠Z = 65°, then ∠Y = 75° too.

∆ABC and ∆XYZ have ∠A = ∠X = 55°, ∠B = ∠Y = 45° and BC = YZ = 6 cm. Are they congruent?

∆ABC has ∠A = 30°, ∠B = 70° and AB = 5 cm. ∆XYZ has ∠X = 30°, ∠Z = 80° and XY = 5 cm. Are they congruent?

Why is AAS really the same as ASA?

• No, they can have different sizes. The side BC = YZ fixes the size. Triangles with the same angles but different sizes would have different sides.
• Yes, by AAS — correct. Yes! Two angles and a side are equal. The third angles are equal too (80°), and BC is then between ∠B and ∠C. ASA applies.
• We cannot say, because BC is not between ∠A and ∠B. It does not need to be. The third angles are equal, so BC is between ∠B and ∠C, which are both known.
• Yes. ∠C = 80° = ∠Z, so ∠A = ∠X, ∠C = ∠Z and AB = XY (AAS). — correct. Yes! ∠C = 180° − 30° − 70° = 80°. So the angles A and C match ∠X and ∠Z, and the side AB = XY.
• No. The angles ∠B and ∠Z differ.. Look at ∠Y. In ∆XYZ, ∠Y = 180° − 30° − 80° = 70°. So ∠B = ∠Y.
• No. They need three equal sides.. They do not. Two angles and a side are enough by AAS.
• Because the two triangles are the same size. That is what we want to prove. The reason is the third angle.
• Because the angles are all 60°. The angles need not be 60°. They only need to add up to 180°.
• Two angles fix the third angle, so we know all three angles and the side lies between two of them. — correct. Yes! The angles add up to 180°, so the third angle is fixed too. Then the side is between two known angles: ASA.