How to find the angles of an isosceles triangle?

Oksana Logunova
Oksana Logunova
April 17, 2013
How to find the angles of an isosceles triangle?

An equilateral triangle is a simple polygon that has three angles and three sides. Before you figure out how to find the angles of an isosceles triangle, you need to know the properties of this geometric figure.

Properties of an isosceles triangle

Consider the properties of an isosceles triangle.

  • In isosceles triangles, the two sides are equal. Its third party is the basis.
  • The angles at the base of such a triangle are equal.
  • The bisector, the median and the height drawn from the angles to the opposite side of the geometric figure are also equal to each other.
  • The bisector, median, and height from the upper angle to the base of an isosceles triangle coincide.
  • If a circle is inscribed inside an isosceles triangle, and also to describe it around such a figure, their centers will lie on one line.
  • The corners at the base can only be sharp.

Thus, if the two angles in the triangle are equal, and its height coincides with the median and the bisector, it is isosceles. This is the main feature of an isosceles triangle.

Now, consider how to find the angles of an isosceles triangle. If such a triangle is also rectangular, then finding its two angles is not difficult, since they will always be equal to 45 degrees, which follows from the properties and signs of an isosceles triangle.

  • Knowing one of the corners, you can always calculate the required. For example, the angle at the base will be denoted by the letter α, the angle at the top of the figure will be denoted by the letter β. Hence, the angle α will be equal to: (π - β) / 2, where π is a constant.
  • Angles can also be calculated using arc sines. To do this, we must describe a circle with a radius around this triangle, which we denote by a large letter R. Then, the angle α = arcsin (a / 2R), and the angle β = arcsin (b / 2R), where a and b are sides of the triangle.

An example of solving the problem

It is necessary to find angles in an isosceles triangle, if it is known that the angle at its base is 15 degrees larger than the angle opposite to the base.

Solution: Denote the opposite angle β, then the angle at the base will be: β + 15. Since the sum in a triangle is always 180 degrees, we find:

β + 2x (β +15) = 180;

β + 2 β + 30 = 180;

3 β = 180-30;

3 β = 150;

β = 50

So, the angle at the base is 50 degrees, which means the other two angles will be equal to 65 degrees each. Now you know the rules of how to find the angles of an isosceles triangle.