ASA condition

The idea

When two angles and the included side of one triangle are equal to two angles and the included side of another, the triangles are congruent; this is the ASA (Angle Side Angle) condition.

Angles and a side

Meera and Rabia measure a triangular frame again. This time one side is 5 cm and the angles at its two ends are 50° and 30°. They think this fixes the frame. Do you?

Do these three measures fix the triangle?

• Yes, only one triangle fits
• No, many triangles fit
• We need the other two sides

The third corner

BC = 5 cm is drawn. The corner A must be on the ray from B at 50°. Slide A along the ray until the angle at C is exactly 30°. How many places work?

Only one place

The ray from B at 50° and the ray from C at 30° meet in one point only. So there is one triangle with these measures. Anyone who draws it gets a triangle that fits on yours.

If two angles and the side between them of one triangle are equal to two angles and the side between them of another triangle, then the triangles are congruent. This is the ASA (Angle Side Angle) condition.

• BC | ∠B | ∠C | ∠A (the rest of 180°)
• 5 cm | 50° | 30° | 100°

Notes

If two angles and the side between them of one triangle are equal to two angles and the side between them of another triangle, then the triangles are congruent. This is the ASA (Angle Side Angle) condition.

Check yourself

∆ABC and ∆XYZ have BC = YZ = 5 cm, ∠B = ∠Y = 50° and ∠C = ∠Z = 30°. Are they congruent?

In ∆ABC, ∠A = 60°, ∠B = 45° and AB = 6 cm. In ∆XYZ, ∠X = 60°, ∠Y = 45° and XY = 6 cm. Which condition shows they are congruent?

In ∆ABC, BC = 5 cm, ∠B = 50° and ∠C = 30°. What is ∠A, in degrees?

Answer: 100

∠A = 180° − 50° − 30° = 100°. Every triangle with these measures has ∠A = 100° as well.

Rabia measures only the angles of a frame: 50°, 30° and 100°. Can she make an exact copy?

• No, we need all three sides. We do not. The side between the two angles already fixes the third corner.
• No, the third angles may differ. They cannot. Both are 180° − 50° − 30° = 100°.
• Yes, by ASA — correct. Yes! Two angles and the side between them are equal.
• SAS. There are two angles and one side here, not two sides.
• SSS. We know only one side. SSS needs all three.
• ASA, because AB is the side between ∠A and ∠B — correct. Yes! AB joins A and B, so it lies between the angles ∠A and ∠B. Angle, Side, Angle.
• Yes, three angles are enough. We saw in Lesson 6 that triangles of different sizes have the same angles.
• Yes, if she knows the colour. Colour does not help. She needs a length.
• Not yet. She also needs one side. — correct. Yes! With the side between two of the angles, ASA fixes the triangle.