Same shape, any size
Two triangles are similar when one is an enlargement of the other: each angle of one is equal to the matching angle of the other, and each side of one is the matching side of the other multiplied by the same number, the scale factor.
That is six facts to check. As with congruence, a few of them are enough to prove all the rest, and there are three tests.
AA: two equal angles
If two angles of one triangle are equal to two angles of another, the triangles are similar. Two angles are enough because the third is forced. The angles of a triangle add to 180°, so if both triangles have angles of 50° and 60°, the third angle is 180° − 50° − 60° = 70° in both.
Here is why equal angles make every side grow by the same factor. Place the small triangle in the corner of the large one, so that one pair of equal angles lie on top of each other and the sides of the two triangles run along the same two lines. The second pair of equal angles is where each triangle's third side meets the base line, so the two third sides cross the base line at the same angle. Two lines that cross one line at the same angle point in the same direction, so they are parallel.
Now enlarge the small triangle from the shared corner, by the scale factor that stretches its base to the large triangle's base. Its third side stays parallel as it moves out, and it ends up through the end of the large base, so it lands exactly on the large triangle's third side. The enlargement is the large triangle.
The small triangle sits in the bottom left corner of the large one and shares its angle there. Its third side runs 2 across and 2 up, parallel to the large triangle's third side, which runs 6 across and 6 up. Its base is 3 and the large base is 9, so every side of the large triangle is 3 times as long.
SSS: three sides in the same ratio
If each side of one triangle is the same multiple of a side of another, the triangles are similar. Pair the sides by size, the shortest with the shortest and the longest with the longest, and divide each pair.
Take sides of 4 cm, 6 cm and 8 cm, and sides of 6 cm, 9 cm and 12 cm: 6 ÷ 4 = 1.5, 9 ÷ 6 = 1.5 and 12 ÷ 8 = 1.5. Every pair gives 1.5, so the triangles are similar with scale factor 1.5. Sides of 6 cm, 9 cm and 13 cm would not be similar to the first triangle, because 13 ÷ 8 = 1.625.
The reason comes from congruence. Enlarge the first triangle by 1.5: its sides become 6, 9 and 12 cm, the same as the second triangle's, so by the SSS congruence test the enlargement is congruent to the second triangle. The second triangle is therefore an enlargement of the first, and its angles are the same.
SAS: two sides in ratio and the angle between them
If two sides of one triangle are the same multiple of two sides of another, and the angles between those sides are equal, the triangles are similar.
One triangle has sides of 5 cm and 8 cm with an angle of 40° between them. Another has sides of 7.5 cm and 12 cm with an angle of 40° between them. 7.5 ÷ 5 = 1.5 and 12 ÷ 8 = 1.5. Enlarge the first triangle by 1.5 and it has sides of 7.5 cm and 12 cm with 40° between them, so by the SAS congruence test it is congruent to the second. The triangles are similar, so the third side of the second triangle is 1.5 times the third side of the first as well.
As with congruence, the equal angle must be the included angle, the one between the two sides.
The small triangle has a base of 2 and an upright side of 3, with a right angle between them. The large one has a base of 4 and an upright side of 6, also with a right angle between them. 4 ÷ 2 = 6 ÷ 3 = 2, so the triangles are similar with scale factor 2.
Corresponding sides come from the angles
In similar triangles, the side opposite an angle in one triangle matches the side opposite the equal angle in the other. So find the equal angles first, and read the sides across from them. Do not match sides by where they sit on the page.
Triangle PQR has angles of 50° at P and 70° at Q. Triangle XYZ has angles of 70° at X and 50° at Z. The third angles are equal too: 180° − 50° − 70° = 60° at R, and 60° at Y. So P matches Z, Q matches X and R matches Y, and triangle PQR is similar to triangle ZXY, with the corners written in matching order.
The side PQ runs between the 50° and 70° angles, and so does ZX, so PQ matches ZX. If PQ = 6 cm and ZX = 9 cm, the scale factor from PQR to ZXY is 9 ÷ 6 = 1.5. The side QR matches XY, so QR = 8 cm gives XY = 8 × 1.5 = 12 cm. Going the other way, divide by 1.5.
Similar and congruent
Congruent triangles are similar triangles with scale factor 1. With a scale factor of 1, the SSS and SAS tests for similarity become the SSS and SAS tests for congruence. The AA test has no scale factor in it at all, which is why equal angles prove similarity and never congruence.
The usual mistakes
Adding instead of multiplying. With scale factor 3, a side of 4 matches 4 × 3 = 12, not 4 + 3 = 7. Similar sides are multiples, so compare them by dividing: 9 − 6 = 3 is not the scale factor from 6 to 9, but 9 ÷ 6 = 1.5 is.
Scaling the wrong side. A side of 4 matches 12 only if the two sides are opposite equal angles. Find the equal angles before pairing the sides.
Using two sides in the same ratio without the angle between them. Two sides alone, or two sides with some other angle, do not prove similarity.
Worked example: The Crossed Legs of an Ironing Board
Question The two legs of an ironing board cross at a pivot X. The board AB rests on the tops of the legs and is parallel to the floor, and the feet C and D stand on the floor. The leg from A runs down through X to the foot D, and the leg from B runs down through X to the foot C. XA = XB = 39 cm, XC = XD = 65 cm, and the feet C and D are 50 cm apart. (a) How far apart are the tops of the legs, A and B? (b) The pivot X is 60 cm above the floor. How high is the board above the floor?
1.AB is parallel to DC, and the leg AD crosses both, so angle XAB is equal to angle XDC because they are alternate angles. Angle AXB is equal to angle DXC because they are vertically opposite angles. Two angles of triangle XAB are equal to two angles of triangle XDC, so the triangles are similar by the AA test.
The board is parallel to the floor, so the alternate angles at A and D are equal, and the angles at X are vertically opposite: triangle XAB is similar to triangle XDC. 2.XA corresponds to XD, so the scale factor from the large triangle to the small one is XAXD = 3965 = 35.
XA corresponds to XD, so the scale factor is 3965 = 35. 3.AB corresponds to DC. (a) The tops of the legs are AB = 35 × 50 = 30 cm apart.
(a) AB corresponds to DC: AB = 35 × 50 = 30 cm. 4.The height of each triangle is measured from X. The height of the large triangle is the 60 cm from X down to the floor, so the height of the small triangle, from X up to the board, is 35 × 60 = 36 cm.
The height of the small triangle, from X up to the board, is 35 × 60 = 36 cm. 5.(b) The board is 60 + 36 = 96 cm above the floor. Check: 3050 = 3660 = 35, the same ratio as 3965.
(b) The board is 60 + 36 = 96 cm above the floor.
Answer: (a) 30 cm; (b) 96 cm
Common mistakes
- Giving 36 cm as the height of the board. That is the height of the small triangle, measured up from the pivot. The board is 36 cm above X, and X is already 60 cm above the floor.
- Using 3965 − 39 = 3926 as the scale factor. The sides that correspond are XA and the whole of XD, from the pivot to the foot, not the difference between them.
More congruence, similarity and circle theorems problems, worked step by step →