What is an acute triangle?

The range that defines the triangle

This is a stronger condition than saying one angle is acute. In fact, every non-degenerate triangle has at least two acute angles, so the classification depends on all three, not on one.

Acute triangles are useful in triangle-center study because all four common centers lie inside the triangle in this case, making the geometry especially clean to visualise.

A triangle with angles 50, 60, and 70 degrees is acute. A triangle with angles 80, 80, and 20 degrees is acute as well. The side lengths do not need to be equal for the angle classification to work.

Do not stop after finding one angle below 90 degrees. Two acute angles are common in many triangles, including right and obtuse triangles. The word acute describes the whole triangle only when every angle passes the test.

The three angle measures still have to add to 180 degrees, so changing one angle changes the room available for the other two.

• An acute triangle has three angles less than 90 degrees.
• All three interior angles must be less than ninety degrees.
• An acute triangle cannot contain a right angle or an obtuse angle.
• In an acute triangle, the centroid, incenter, circumcenter, and orthocenter all lie inside the figure.

Where acute triangles help

• Use acute-triangle classification when sorting triangles by angle measure.
• Use it in center-location discussions where the position of special points matters.
• Use it before applying theorems that distinguish acute, right, and obtuse cases.

Three acute-triangle checks

• Do not classify a triangle as acute after checking only one or two of its angles.
• Do not confuse an acute triangle with a triangle that merely contains some acute angles.
• Do not let a rotated drawing hide the fact that one angle has reached or exceeded ninety degrees.