SAS condition

The idea

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

How many measures do we need?

Meera has the two sides of the frame, 6 cm and 5 cm, and the 30° angle between them. Rabia says this is enough to make an exact copy. Is she right?

Do two sides and the angle between them fix the triangle?

• No, we need all three sides
• Yes, the triangle is fixed
• Only if the angle is 90°

Swing the arm

Two arms, AB = 6 cm and AC = 5 cm, are joined at the hinge A. Swing the arm AC and watch the third side BC. Try some angles, then set the angle to 30°. Then slide each classmate's triangle onto yours.

The SAS rule

Other angles gave other triangles, with other lengths of BC. At 30° the triangle was fixed: BC came to about 3.0 cm, and every classmate's triangle, turned or flipped, fitted on mine.

If two sides and the angle between them of one triangle are equal to two sides and the angle between them of another triangle, the triangles are congruent. This is the SAS (Side Angle Side) condition.

• Angle at A | Side BC
• 10° | 1.4 cm
• 30° | 3.0 cm
• 70° | 6.4 cm
• 150° | 10.6 cm

Notes

If two sides and the angle between them of one triangle are equal to two sides and the angle between them of another triangle, the triangles are congruent. This is the SAS (Side Angle Side) condition.

Check yourself

In ∆ABC, AB = 6 cm, AC = 5 cm and ∠A = 30°. In ∆XYZ, XY = 6 cm, XZ = 5 cm and ∠X = 30°. What can we say?

∆ABC has AB = 7 cm, BC = 4 cm and ∠B = 50°. ∆PQR has PQ = 7 cm, QR = 4 cm and ∠Q = 50°. Which rule shows they are congruent?

∆ABC has AB = 6 cm, AC = 5 cm and ∠C = 30°. ∆XYZ has XY = 6 cm, XZ = 5 cm and ∠X = 30°. Is SAS true here?

Two sides of a triangle are 6 cm and 5 cm, and the angle between them is 30°. A classmate draws the same triangle in her book. Its third side is about how many cm? (Use the table in the box above.)

Answer: 3

Both triangles are congruent by SAS, so their third sides are equal: BC is about 3.0 cm.

• We cannot say. We need the third side too.. We do not. Two sides and the angle between them already fix the third side.
• They are congruent only if they point the same way. Turning or flipping does not matter. Congruent triangles can be turned or flipped.
• They are congruent by SAS — correct. Yes! Two sides and the angle between them are equal.
• SSS. SSS needs all three pairs of sides. Here we know two sides and an angle.
• We cannot tell. We can. The angle is between the two sides in both triangles.
• SAS — correct. Yes! ∠B is between AB and BC, and ∠Q is between PQ and QR. Side, Angle, Side.
• Yes, 6, 5 and 30° are all equal. Look at where the angle sits. In ∆ABC the 30° angle is at C, not between AB and AC.
• No. In ∆ABC the angle is not between the two sides. — correct. Yes! SAS needs the angle between the two sides. ∠C is not between AB and AC. So SAS does not apply.
• Yes, because both triangles have a 30° angle. The angle must be between the two equal sides. ∠C is not between AB and AC.