Three sides, one triangle

The idea

If two triangles have the same three sidelengths, then they are congruent; this is the SSS (Side Side Side) condition for congruence.

Are the sides enough?

Meera and Rabia must make a cardboard triangle exactly like the school frame. They measure its sides: 40 cm, 60 cm and 80 cm. Rabia gets out her protractor, but Meera says: “We do not need the angles!”

Is Meera right? Can the three side lengths alone give the exact triangle?

• Yes, the three sides are enough
• No, we also need the angles
• We cannot tell

How many triangles?

Use 4, 6 and 8 cm so it fits the page. AB = 6 cm is drawn. The third corner must be 4 cm from A, so it slides on this circle. Where is it also 8 cm from B?

Two places, one triangle

There were exactly two places, E and F, one above AB and one below it. They are mirror images in AB, so ∆ABE and ∆ABF fit exactly: they are congruent.

If two triangles have the same three sidelengths, then they are congruent. This is the SSS (Side Side Side) condition for congruence.

• The three sides are | The triangle
• 4 cm, 6 cm, 8 cm | always the same one, up to a flip
• the same 3 lengths again | congruent

Notes

If two triangles have the same three sidelengths, then they are congruent. This is the SSS (Side Side Side) condition for congruence.

Check yourself

Anil and Bela each draw a triangle with sides 5 cm, 6 cm and 7 cm. Must their two triangles be congruent?

Triangle 1 has sides 4 cm, 6 cm and 8 cm. Triangle 2 has sides 8 cm, 4 cm and 6 cm, written in a different order. Are they congruent?

Triangle PQR has sides of 4 cm, 6 cm and 8 cm. Triangle XYZ has sides of 8 cm, 4 cm and x cm. Both are congruent by SSS. What is x, in cm?

Answer: 6

The three lengths of PQR are 4, 6 and 8. XYZ has 8 and 4, so the last side is 6 cm.

Triangle ABC has sides 5 cm, 6 cm and 7 cm. Triangle DEF has sides 5 cm, 6 cm and 8 cm. Are they congruent?

• No. We must also know the angles.. No need. The three sides fix the triangle, as Meera said. The corner has only two places, which are mirror images.
• Yes. The same three sidelengths always give congruent triangles. — correct. Yes! This is the SSS condition. One may be flipped over, but they still fit.
• No. Triangles with the same sides can differ in shape.. They cannot. The third corner has to be at the right distance from both ends of the base, and that leaves only two mirror-image places.
• Yes. They have the same three lengths, so SSS holds. — correct. Yes! The order in which we write the lengths does not matter. Each length appears once in each triangle.
• No. The lengths are written in a different order.. Writing the lengths in a different order does not change the triangle. They are the same three lengths.
• We cannot say without measuring the angles.. We can say. SSS tells us that equal sides are enough.
• No. The third sides, 7 cm and 8 cm, are different. — correct. Yes! SSS needs all three lengths to match. Two sides alone are not enough, and here the third side differs.
• Yes. Two of their sides are the same.. Two equal sides are not enough for SSS. All three sidelengths must be the same, and 7 cm is not 8 cm.
• Yes. Both can be flipped to fit.. Flipping never changes a length. The 7 cm side would still not match the 8 cm side.