Relationship between angles and sides of a triangle

Have you noticed that there is a relationship between the lengths of sides and the size of angles in triangles?

For example, if all the sides of a triangle are the same size then so are all the angles and vice-versa. The triangle is an equilateral triangle.

If two or the sides are equal then so are two of the angles and vice-versa. The triangle is an isosceles triangle.

If no sides are equal then no angles are. The triangle is a scalene triangle.

There is another important relationship between sides and angles.

In two similar triangles, the angles are the same in both triangles but the lengths of the sides are different. However the ratio of the lengths in both triangles are the same.

To explain what I mean, I’ll use an example.

Look at the two triangles above.

They are similar.

All the corresponding angles are the same size. All the corresponding sides are unequal.

In triangle ABC the hypotenuse is 15 units, the hypotenuse in triangle DEF is 10.

15 ÷ 10 = 1.5

The base of the triangle ABC, adjacent to angle 53.1 is 9. The base of triangle DEF, adjacent to angle 53.1 is 6.

9 ÷ 6 = 1.5

The length of the side opposite angle 53.1 in triangle ABC is 12. The length of the side opposite angle 53.1 in triangle DEF is 8.

12 ÷ 8 = 1.5

This is the same as saying the ratio of the length of the hypotenuse in triangle ABC to the length of the hypotenuse in triangle DEF is 1.5. The ratio of the lengths of the sides adjacent and opposite angle 53.1 in triangle ABC to triangle DEF are also 1.5.

This is the same as saying that the ratio of the lengths of the sides in triangle ABC to the corresponding sides in triangle DEF is 1.5.

The ratios of the corresponding sides in similar triangles are all the same.