If you have a protractor or an angle-finder tool, you can use that to check the angle. To deduce angles using mathematics, we use the trigonometric functions sine, cosine and tangent, basic triangular geometry and some algebra.

The 180 degree rule of triangles

In any given triangle, all three angles must add up to 180 degrees. Thus if we know any two of the angles, we can deduce the third by subtracting the sum of those two angles from 180.

If we only know one of the angles, then we need to know the lengths of at least two sides to determine the other angles.

Labelling triangles

It is useful to label triangles in a standard fashion. We normally label the three sides using lower case, a, band c. The angles opposite these sides are labelled in upper case, A, B and C, such that angle A is opposite side a and angle B is opposite side b.

Sine rule

The sine rule is used whenever we know the lengths of any two sides and at least one opposing angle, or we know two angles and at least one opposing side. The sine rule states that:

a/sin(A) = b/sin(B) = c/sin(C)

Cosine rule

The cosine rule is used when we know at least two sides and only one angle, where the known angle is not opposite a known side. The cosine rule states that:

cos(A) = (b2 + c2 - a2) / 2bc

Thus we can determine the length of the third side using:

a2 = b2 + c2 - 2bc cos(A)

Knowing all three sides allows us to determine either of the remaining angles using the sine rule followed by the 180 rule to determine the final angle.

Right-angled triangles and Pythagoras' theorem

A right-angled triangle is any triangle where one of the angles is exactly 90 degrees. In order to discover the other two angles we must know the lengths of at least two sides. The side opposite the right-angle is always known as the hypotenuse (and is always the longest side).

Pythagoras' Theorem tells us that, for any right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides. Thus for any right-angled triangle where a is the hypotenuse:

a2 = b2 + c2

Knowing all three sides allows us to determine the two missing angles using the sine rule. However, in right angled triangles, we can use a simpler method. If we want to discover the angle of B, where a is the hypotenuse, then c becomes the adjacent side and b is the opposite side. Thus any of the following can be used to determine the angle of B:

sin(B) = opposite/hypotenuse

cos(B) = adjacent/hypotenuse

tan(B) = opposite/adjacent

In other words, where angle A is 90 degrees:

sin(B) = b/a

cos(B) = c/a

tan(B) = b/c

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