Matching parts

The idea

When two congruent triangles are made to overlap, their corresponding vertices, sides and angles fit exactly over each other, so corresponding sides are equal and corresponding angles are equal.

Which corner goes where?

Two triangles, ABC and XYZ, are congruent. We lay one over the other so that they fit exactly.

Can we lay ABC on XYZ in any way we like, or must the corners match up in a particular way?

• Only equal sides can lie on each other, so the corners must match up in a particular way
• Any corner can go on any corner
• Only the flat sides matter

Find the partners

The paper copy of ∆ABC has to cover ∆XYZ. When it fits, tap each corner of the paper: which corner of ∆XYZ is under it?

Partners fit over each other

A landed on X, B on Y, C on Z. The sides that lay on each other are equal, and so are the angles at the corners that lay on each other.

When two congruent triangles are made to overlap, their corresponding vertices, sides and angles fit exactly over each other. So corresponding sides are equal and corresponding angles are equal.

• Corresponding | Pairs
• vertices | A and X, B and Y, C and Z
• sides | AB and XY, BC and YZ, CA and ZX
• angles | ∠A and ∠X, ∠B and ∠Y, ∠C and ∠Z

Notes

When two congruent triangles are made to overlap, their corresponding vertices, sides and angles fit exactly over each other. So corresponding sides are equal and corresponding angles are equal.

Check yourself

Look at the tick marks: sides with the same number of ticks are equal. Which angle of ∆XYZ is equal to ∠B of ∆ABC?

∆ABC and ∆XYZ are congruent, with A on X, B on Y and C on Z. In ∆ABC, ∠A = 70° and ∠B = 60°. How big is ∠Z, in degrees?

Answer: 50

C lies on Z, so ∠Z = ∠C. The angles of ABC add up to 180°: ∠C = 180° − 70° − 60° = 50°. So ∠Z = 50°.

∆ABC and ∆PQR are congruent, with A on P, B on Q and C on R. Which statement is true?

∆ABC ≅ ∆PQR with A on P, B on Q, C on R. In ∆ABC, AB = 8 cm, BC = 11 cm and CA = 9 cm. How long is PR, in cm?

Answer: 9

P is where A lay and R is where C lay, so PR lay on AC. PR = CA = 9 cm.

• ∠X. ∠X sits between the one-tick side XY and the three-tick side XZ. That matches ∠A, not ∠B.
• ∠Y — correct. Yes! ∠B sits between the one-tick side AB and the two-tick side BC. In ∆XYZ that is ∠Y, between XY and YZ.
• ∠Z. ∠Z sits between the two-tick side YZ and the three-tick side XZ. That matches ∠C, not ∠B.
• AB = QR, BC = RP and CA = PQ. AB lies on PQ, not on QR. Only the sides that lie on each other are equal.
• AB = PQ, BC = QR and CA = RP — correct. Yes! Each side is equal to the side that lies on it. Partners are equal.
• All six sides are equal to each other. Not at all. In ∆ABC itself the sides can all be different. Only the partners are equal.