# The largest angle of an acute-angled triangle is five times the smallest. Find the angles

**The largest angle of an acute-angled triangle is five times the smallest. Find the angles of this triangle, if you know that they are all expressed as an integer number of degrees.**

Let’s designate a smaller angle x, then the larger one will be 5 * x.

The angles in a triangle add up to 180 °. So the average angle is:

180 – (5 * x + x) = 180 – 6 * x.

By condition, our triangle is acute-angled. This means that the largest angle is less than 90 °.

Let’s compose 1 inequality:

5 * x <90 °;

x <18 °.

The middle angle should be less than the large one. Let’s compose 2 inequalities:

180 – 6 * x <5 * x;

11 * x> 180 °.

x> 16.36 °.

16.36 ° <x <18 °.

x = 17 °.

5 * x = 17 * 5 = 85 °.

180 – (17 + 85) = 78 °.

Answer: The angles of the triangle are 17 °, 78 ° and 85 °.