Geometry is a fascinating realm of mathematics, filled with intriguing concepts and angles that shape our understanding of space and form. In this article, we embark on an indepth exploration of acute angles, covering their definition, degree measurement, visual representations, mathematical formulas, practical examples, properties within triangles, realworld applications, and the crucial distinctions between acute and obtuse angles.
Table of Contents
Acute Angle Definition:
An acute angle is an angle that measures less than 90 degrees. In other words, it is an angle whose magnitude is smaller than a right angle. For example, an angle measuring 20 degrees, 75 degrees, or 89 degrees would all be acute angles.
Acute Angle Degree:
The measure of an acute angle is always less than 90 degrees. The degree measure of an acute angle could be any value between 0 and 90 degrees, exclusive.
Acute Angle Images:
Acute Angle Formula:
In the context of an acute triangle, the relationships among the lengths of its sides are governed by the following conditions:
 The sum of the squares of the two shorter sides, denoted as a and b, is greater than the square of the longest side c: a^{2} + b^{2} > c^{2}
.  The sum of the squares of the second shortest side, b, and the longest side, c, is greater than the square of the shortest side, a: b^{2} + c^{2} > a^{2}
.  The sum of the squares of the shortest side, a, and the second longest side, c, is greater than the square of the remaining side, b: c^{2} + a^{2} > b^{2}
Acute Angle Examples:
Examples of acute angles could include:
 A 30degree angle.
 A 60degree angle.
 Any angle with a measure between 0 and 90 degrees.
Acute Angle Triangle:
An acuteangled triangle is a triangle in which all three angles are acute angles, meaning each angle is less than 90 degrees.
Properties of Acute Triangle:

 Properties of an acute triangle include:
 All three angles are acute.
 The sum of the interior angles is less than 180 degrees.
 No angle is a right angle or obtuse angle.
 Properties of an acute triangle include:
Acute Angle vs Obtuse Angle
Characteristic  Acute Angle  Obtuse Angle 

Definition  An angle that measures greater than 0 degrees  An angle that measures greater than 90 degrees 
Measure Range  Between 0 degrees and less than 90 degrees  Between 90 degrees and less than 180 degrees 
Symbolic Representation  ∠ABC, where the measure of ∠ABC < 90 degrees  ∠PQR, where the measure of ∠PQR > 90 degrees 
Examples  30 degrees, 45 degrees, 60 degrees  100 degrees, 120 degrees, 150 degrees 
Triangle Classification  All angles in an acuteangled triangle are acute  One angle in an obtuseangled triangle is obtuse 
Trigonometric Functions  Trigonometric functions are defined for acute angles  Trigonometric functions are defined for obtuse angles 