Identical, however it is turned
Two shapes are congruent when they are exactly the same shape and exactly the same size. One of them may be turned round, or flipped over like a page, but if it were cut out and laid on the other, it would cover it exactly: every side on a side of the same length, and every angle on an equal angle.
The two triangles below are congruent. Each has a side of 3 and a side of 4 with a right angle between them. The one on the right is the one on the left turned a quarter turn.
On the left, the sides at the right angle run 3 across and 4 up. On the right, they run 4 across and 3 down. Turned a quarter turn, the left triangle covers the right one exactly.
Six parts, and three are enough
A triangle has six parts: three sides and three angles. To prove that two triangles are congruent, you do not have to check all six. Three are enough, if they are the right three, because the right three leave only one triangle that can be drawn. Any two triangles that share those three facts are then copies of each other, and their other three parts are equal too.
There are four sets of three that work, and each is named by its letters: S for a side and A for an angle, in the order they come round the triangle.
SSS: three sides
Take three sides of 7 cm, 5 cm and 6 cm. Rule the 7 cm side first. The third corner must be 5 cm from one end of it and 6 cm from the other, so draw an arc of 5 cm from one end and an arc of 6 cm from the other. The arcs cross at one point above the side, and that point is the third corner. At no step was there a choice.
The arcs also cross at one point below the side, but the triangle built there is the first one flipped over, so it is congruent to it. So every triangle with sides of 7 cm, 5 cm and 6 cm is a copy of every other, and all of them have the same three angles. This is the side-side-side test, SSS.
All three sides are given. Only one triangle has these three lengths, so its angles are fixed as well.
SAS: two sides and the angle between them
Draw the angle first: two arms leaving one corner. Measure the first side along one arm and the second side along the other. The far ends of the two sides are now fixed points, and the third side is the straight line joining them. There is only one straight line between two points, so the whole triangle is fixed.
The angle must be the one between the two sides, called the included angle. The name side-angle-side, SAS, says so: going round the triangle, the angle comes between the two sides.
Sides of 4 and 5 with an angle of 78° between them. The third side can only run straight from the far end of one to the far end of the other.
ASA: two angles and a side
Draw the side. At each end of it, draw a line at its given angle. Two lines that are not parallel cross at exactly one point, and that point is the third corner. The two angles fix the shape of the triangle and the side between them fixes its size. This is angle-side-angle, ASA.
The side does not have to be the one between the two angles. The angles of a triangle add to 180°, so two angles give the third: with angles of 62° and 54°, the third is 180° − 62° − 54° = 64°. Now the known side lies between two known angles, whichever side it was. This form is often written AAS.
A side of 5.1 with angles of 62° and 54° at its two ends. The lines drawn from its ends at those angles cross at one point, the third corner.
RHS: right angle, hypotenuse, side
In a right-angled triangle, the hypotenuse is the side opposite the right angle, and it is the longest side. If two right-angled triangles have equal hypotenuses and one other pair of equal sides, Pythagoras' theorem fixes the third side of each. With a hypotenuse of 5 cm and a side of 3 cm, the other side is cm, in both triangles.
All three sides then match, so the triangles are congruent by SSS. This is the right angle-hypotenuse-side test, RHS.
A right-angled triangle with a hypotenuse of 5 and one side of 3. The other side can only be 4.
Two sides and an angle not between them
Two sides and an angle that is not between them do not always fix a triangle, so SSA is not a test. The figure shows why.
The bottom left corner is A, and the top corner is B. AB runs 4 across and 4 up, along the diagonals of the grid squares, so the angle at A is 45°, and the side AB is fixed. The third side is to be 5 units long, from B to a corner on the bottom line. From B, a length of 5 reaches the bottom line at two points: C, 3 units to the left of the point below B, and D, 3 units to the right, because .
So triangle ABC and triangle ABD both have the angle of 45° at A, the same side AB, and a side of 5 from B. They are not congruent: AC is 1 unit long and AD is 7. The angle at C is obtuse and the angle at D is acute.
SSA does fix the triangle when the side opposite the given angle is the longer of the two given sides: then the second crossing point lands on the far side of A, where it makes no triangle. RHS is exactly that case, because the right angle is opposite the hypotenuse, the longest side.
A is the bottom left corner and B the top. From B, one side of 5 goes 3 across to the left and 4 down, to C, and another goes 3 across to the right and 4 down, to D. Triangle ABC is the small one inside triangle ABD.
Three angles are not a test
Three equal angles fix the shape but not the size. A triangle with angles of 50°, 60° and 70° can be drawn with a base of 1 cm or of 1 m. Triangles whose angles are equal are similar: the same shape, but not necessarily the same size. So AAA never proves congruence.
Matching the letters in order
Writing "triangle ABC is congruent to triangle PQR" says which corners match, by their order: A matches P, B matches Q and C matches R. Everything else follows from that order. The side AB is equal to PQ, BC to QR and AC to PR, and the angle at A is equal to the angle at P.
Here is an example. In triangle ABC, AB = 6 cm, BC = 7 cm and the angle at B is 50°. In triangle XYZ, YZ = 6 cm, ZX = 7 cm and the angle at Z is 50°. The 50° angles are between the two given sides in both, so the triangles are congruent by SAS, with B matching Z. The side of 6 cm from B ends at A and the side of 6 cm from Z ends at Y, so A matches Y. The sides of 7 cm end at C and at X, so C matches X.
So the statement is "triangle ABC is congruent to triangle YZX". It then tells you that AC = YX, the third sides, without measuring either one. Writing triangle XYZ in that place would claim that AB = XY, which is not true.
Proving a fact with congruence
Congruence is how many facts about shapes are proved. Take an isosceles triangle ABC, with AB = AC, and join the top corner A to M, the midpoint of the base BC.
Triangles ABM and ACM have AB = AC, BM = CM because M is the midpoint, and the side AM in common. By SSS, triangle ABM is congruent to triangle ACM. So the angle at B is equal to the angle at C: the base angles of an isosceles triangle are equal. The two angles at M are equal too, and together they make a straight line of 180°, so each is 90°.
An isosceles triangle, cut from its top corner to the midpoint of its base. The two halves have three pairs of equal sides, so they are congruent.
The usual mistakes
Taking three equal angles as proof. AAA shows the triangles have the same shape; they can still be different sizes.
Using an angle that is not between the two sides. Two sides and the angle between them is SAS. Two sides and another angle is SSA, which can fit two different triangles.
Matching the corners out of order. Find the pairs of equal parts first, then write the two triangles with matching corners in the same positions.
Before the application
In the application below, SAS settles the third edge of the second sail. Its last step checks that length with the cosine rule, a formula from trigonometry that calculates the third side of a triangle from two sides and the included angle. The answer does not depend on it: congruence alone says the edge is 7 m.
Worked example: Two Triangular Sails Cut to the Same Pattern
Question A sailmaker has cut two triangular sails. On sail ABC, the edge AB is 8 m long, the edge BC is 5 m long and the angle between them, angle ABC, is 60°. The third edge AC has been measured as 7 m. On sail PQR, the edge PQ is 8 m long, the edge QR is 5 m long and angle PQR is 60°, but the edge PR has not been measured. (a) How long is PR? (b) A tape is to be sewn along all three edges of sail PQR. How many meters of tape are needed?
1.Compare the two sails. AB and PQ are both 8 m, BC and QR are both 5 m, and the angles between these pairs of sides, angle ABC and angle PQR, are both 60°.
The two sails share two sides and the angle between them: 8 m, 5 m and 60°. 2.Two sides and the angle between them are equal in the two triangles, so triangle ABC is congruent to triangle PQR by the SAS test. The corners match in the order A to P, B to Q and C to R.
Two sides and the included angle: triangle ABC is congruent to triangle PQR by SAS. 3.In congruent triangles every matching side is equal. AC is opposite the 60° angle at B, and PR is opposite the 60° angle at Q, so PR matches AC. (a) PR = AC = 7 m.
(a) PR matches AC, the side opposite the 60° angle, so PR = 7 m. 4.The tape runs along PQ, QR and PR. (b) The tape needed is 8 + 5 + 7 = 20 m. Check: the cosine rule on sail ABC gives AC2 = 82 + 52 − 2 × 8 × 5 × cos 60° = 64 + 25 − 40 = 49, so AC = 7 m, as measured.
(b) The tape runs round all three edges of PQR: 8 + 5 + 7 = 20 m.
Answer: (a) 7 m; (b) 20 m
Common mistakes
- Saying the sails cannot be compared because only two sides of sail PQR are known. Two sides and the angle between them fix a triangle completely, so the third side is already decided.
- Matching PR with AB because both are edges that meet at the 60° angle. PR is the edge opposite the 60° angle, so it matches AC, the edge opposite the 60° angle in the first sail.
More congruence, similarity and circle theorems problems, worked step by step →