Acute triangle

High School

Definition

A triangle that contains three acute angles. Which is, all three interior angles must be less than 90°. There are three major types of acute triangles. The first is equilateral triangle. It has three equal angles all measuring at 60° and three equal sides. The second is acute isocelese triangle. It has two equal angles and two equal sides. The third type is acute scalene triangle. It is when there are no angles and no equal sides.

Worked examples

\(\triangle ABC\) with angles \(50^\circ\), \(60^\circ\), \(70^\circ\)

All three angles are less than \(90^\circ\), so this is an acute triangle.

Equilateral triangle: \(60^\circ\), \(60^\circ\), \(60^\circ\)

Three equal angles that are all acute make this a special case of an acute triangle.

Acute isosceles: \(70^\circ\), \(70^\circ\), \(40^\circ\)

Two equal acute angles with a third different acute angle — all under \(90^\circ\).

Common mistakes

A triangle with angles \(30^\circ\), \(60^\circ\), \(90^\circ\) is acute → That triangle is right, not acute An acute triangle requires all three angles to be strictly less than \(90^\circ\).
An acute triangle must have equal sides → An acute triangle can be scalene with no equal sides Acute describes the angles only; sides can be all different, two equal, or three equal.
A triangle with one \(85^\circ\) angle and one \(50^\circ\) angle is acute → Check all three: \(85^\circ + 50^\circ + 45^\circ = 180^\circ\), all \(< 90^\circ\), so yes Always verify that all three angles sum to \(180^\circ\) and each is under \(90^\circ\).

Where you'll use it next

You'll classify acute triangles when solving for unknown angles, applying the Law of Sines and Cosines, and analyzing triangle congruence and similarity in geometry and trigonometry.

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Mastering Triangle Classification: Your Key to Geometry Success

Geometry

Unlock the world of geometry by mastering triangle classification. Learn to identify and understand various triangle types, their properties, and real-world applications. Boost your problem-solving skills today!

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