Corresponding sides are the pairs of sides that sit in the same position in two similar shapes: each one is opposite the same angle, or between the same two angles. Find these pairs before you write any ratio, because a wrong pair makes every later line of working wrong.
This skill opens the module on similarity, congruence and scale, and every scale factor question depends on it.
How do you find the corresponding sides?
Start from the angles, not the letters. Two angles that are equal tell you which corners match. The side opposite a corner then matches the side opposite the corresponding corner.
A useful habit is to list the corresponding corners first, for example “A ↔ E, B ↔ F, C ↔ D”. Then read each side off that list: side AB uses corners A and B, so it matches side EF.
Two quick checks help. The longest side matches the longest side, and the shortest matches the shortest.
Worked example
Triangle ABC has angle A = 40°, angle B = 75° and angle C = 65°. Its sides are AB = 10 cm, BC = 8 cm and CA = 6 cm.
Triangle DEF has angle D = 65°, angle E = 40° and angle F = 75°. Side EF = 15 cm. Find FD and DE.
Step 1, match the corners by angle. 40° is at A and at E, so A ↔ E. 75° is at B and at F, so B ↔ F. 65° is at C and at D, so C ↔ D.
Step 2, match the sides. AB uses A and B, so it matches EF. BC uses B and C, so it matches FD. CA uses C and A, so it matches DE.
Step 3, find the scale factor from the pair you know: EF ÷ AB = 15 ÷ 10 = 1.5.
Step 4, use it. FD = 8 × 1.5 = 12 cm. DE = 6 × 1.5 = 9 cm.
Step 5, check. The longest sides are 10 and 15, the shortest are 6 and 9, and 10 × 1.5 = 15. The pairs are consistent.
The mistake to watch for
A common slip is to match sides by letter order, so that AB is paired with DE because both start the name in the same place.
Mistaken working: AB ↔ DE, so the scale factor is DE ÷ AB. The student has no value for DE, gets stuck, or pairs 15 with 6 and writes 15 ÷ 6 = 2.5.
The pairing ignores the angles, so the ratio is wrong even though the arithmetic is correct.
The correction is to match the equal angles first. Here 15 cm sits opposite the 65° angle at D, and 10 cm sits opposite the 65° angle at C, so the correct pair is 15 ↔ 10. The longest side must match the longest side, which is another quick way to catch this.
Check yourself
Work these out, then open each answer.
1. In triangle PQR, P = 50°, Q = 60°, R = 70°. In triangle STU, S = 70°, T = 50°, U = 60°. Which side of STU corresponds to PQ?
Show answer
P = 50° matches T, and Q = 60° matches U. So PQ corresponds to TU. As a check, R = 70° matches S, so PR corresponds to TS and QR corresponds to US.
2. One triangle has sides 5, 7 and 9. Another has sides 10, 14 and 20. Are they similar?
Show answer
Match shortest to shortest, middle to middle, longest to longest. The ratios are 10 ÷ 5 = 2, 14 ÷ 7 = 2 and 20 ÷ 9 ≈ 2.22. The ratios are not all equal, so the triangles are not similar.
3. In triangle ABC, point D lies on AB and point E lies on AC, with DE parallel to BC. Which side of triangle ABC corresponds to DE, and which side corresponds to AD?
Show answer
Angle ADE equals angle ABC because DE is parallel to BC, so D matches B. Angle AED equals angle ACB, so E matches C. Angle A is shared. So DE corresponds to BC, and AD corresponds to AB.
Where this leads next
With the pairs identified, you are ready to use a linear scale factor to find missing lengths. The non-calculator working trainer is useful for practising clean, checkable ratio working.
Some students understand each step here but still mis-pair sides when a diagram is rotated or overlapping. That is the kind of pattern our teachers look for in online one-to-one Mathematics tuition.