# Acute triangle

An acute triangle is a triangle with three acute angles, which are angles measuring less than 90°.

## Properties of acute triangles

- Whenever a triangle is classified as acute, all of its interior angles have a measure between 0 and 90 degrees.
- The side opposite the largest angle of a triangle is the longest side of the triangle. The greater the measure of an angle opposite a side, the longer the side. Conversely, the longer the side the greater the measure of the opposing angle. In the triangle above, since DF = 10 is the longest side of the triangle, the angle opposite DF, ∠E, must be the greatest angle in the triangle. (∠E = 83°. ∠D and ∠F are 51° and 46° respectively).
- The orthocenter is the point where all three altitudes of the triangle intersect. The orthocenter for an acute triangle is located inside of the triangle, as shown in the figure below where O is the orthocenter of triangle ABC.

## Using the Pythagorean Theorem to find acute triangles

When the lengths of the sides of a triangle are known, the Pythagorean Theorem can be used to determine whether or not the triangle is an acute triangle.

For a right triangle with a hypotenuse of length c and leg lengths a and b, the Pythagorean Theorem states:

a^{2} + b^{2} = c^{2}

On the other hand, in a triangle where a^{2} + b^{2} > c^{2}, if side c is also the longest side, the triangle is an acute triangle.

Example:

Is triangle ABC an acute triangle?

Side AB above is the longest side of triangle ABC. To check if ABC is an acute triangle, let c = 13, a = 10 and b = 9:

a^{2} + b^{2} = 10^{2} + 9^{2} = 181

c^{2} = 13^{2} = 169

181 > 169

Therefore triangle ABC is an acute triangle.