Writing ≅ in order

The idea

Congruence is written ΔABC ≅ ΔXYZ, listing vertices in corresponding order: first vertex corresponds to first, second to second, third to third, so the order must match the correspondence.

Does the order matter?

Two triangles are congruent. A lies on X, B on Y and C on Z. We write ∆ABC ≅ ∆XYZ.

Could we also write ∆ACB ≅ ∆XYZ?

• Yes, it is the same triangle
• No, the order of the letters matters
• Yes, but only if the triangles are turned

Write it so it fits

The toy checks the three sides for the order you choose. Tap letters into the slots. Find three different ways that are correct.

First with first, second with second

Every correct way matched the first letter to the first, the second to the second and the third to the third. The wrong way mixed them up.

Congruence is written ∆ABC ≅ ∆XYZ, listing the vertices in corresponding order: the first vertex corresponds to the first, the second to the second, the third to the third. So the order must match the correspondence.

• Written | Correct?
• ∆ABC ≅ ∆XYZ | Yes
• ∆ACB ≅ ∆XZY | Yes: C and Z swap places together
• ∆ACB ≅ ∆XYZ | No: C is 2nd on the left, but Y is 2nd on the right

Notes

Congruence is written ∆ABC ≅ ∆XYZ, listing the vertices in corresponding order: the first vertex corresponds to the first, the second to the second, the third to the third. So the order must match the correspondence.

Check yourself

Which is a correct way to write the congruence of ∆ABC and ∆XYZ, where A lies on X, B on Y and C on Z?

The same number of ticks marks equal sides. Which statement is correct?

∆DEF ≅ ∆MNP, with D on M, E on N and F on P. In ∆DEF, ∠D = 65° and ∠E = 55°. How big is ∠P, in degrees?

Answer: 60

P is the third letter, so P is partner to F. ∠F = 180° − 65° − 55° = 60°. So ∠P = 60°.

Anil writes ∆ABC ≅ ∆XYZ. What does this tell us about the sides AB and XY?

• ∆BCA ≅ ∆XYZ. B is first on one side and X is first on the other, but B lies on Y, not X. The order must match.
• ∆BCA ≅ ∆YZX — correct. Yes! B and Y are both first, C and Z second, A and X third. The partners are in the same places.
• ∆CAB ≅ ∆YXZ. C is first and Y is first, but C lies on Z, not Y. The partners must be in the same places.
• ∆PQR ≅ ∆LMN. This puts P first and L first, but PQ has one tick and LM has two. The first letters are not partners.
• ∆PQR ≅ ∆MNL. This puts P first and M first, but PQ has one tick and MN has three. The first letters are not partners.
• ∆PQR ≅ ∆NLM — correct. Yes! PQ (1 tick) is NL, QR (2 ticks) is LM and PR (3 ticks) is NM. So P is on N, Q on L, R on M.
• AB = XY, because the first two letters of each triangle are partners — correct. Yes! The first two letters of ∆ABC are A, B and of ∆XYZ are X, Y. Partner sides are equal.
• AB = YZ, because both come second. The side AB is made of the 1st and 2nd letters. Its partner is the side made of the 1st and 2nd letters of ∆XYZ: XY.
• Nothing. We only know the triangles are the same size. The statement tells more. It says which sides and angles are partners.