Angle is the geometric concept you learn after the line, line segment and ray. Indeed the idea of angle opens up and leads to more ideas and concepts and thus contributes to the expansion of the subject. An angle is formed by two intersecting rays at the point of intersection. The rays are called the arms or sides of the angle and the point of intersection is called the vertex of the angle.

## Acute Angle Definition

An acute angle is an angle whose degree measure is greater than 0 but less than 90.
In general angle if angle A is acute its degree measure < 90º.

In the above diagram angle 1 is seen smaller than the right angle marked by a square sign. Hence it is an acute angle.

Similarly, the angle marked as 2 is called the complementary angle of angle 1. The measures of angles 1 and 2 add up to 90º.
Hence the complementary angle of an acute angle is also acute as it measures less than 90º.
In the adjoining diagram angle 1 is formed at the intersection of the rays AB and AC.
AB and AC are called the arms or sides of the angles. A is the vertex where the angle is formed.
The angle is denoted in any of the following manner.
∠A
∠BAC
∠CAB or ∠1.

Degree is the unit used to represent an angle measure which is denoted by the symbol º. The measure of the angle around a circle is 360º.

## Acute Angle Examples

Example 1:
The sum of the angles in a triangle is 180º. Hence any triangle should contain at least two acute angles.
When all the three angles of the triangle are acute, the triangle is classified as an acute angled triangle.

Example 2:
If an angle is acute, then its complementary is also acute, but is supplementary is obtuse.
For example, consider the angle measure 50º. Its complementary = 90 -50 = 40 < 90 and hence acuted.
The supplementary of 50 is 180 - 50 = 130 > 90, which is obtuse.

Example 3:
 A parallelogram whose adjacent sides are not perpendicular has one pair of acute angles equal measure and placed opposite to each other.

Example 4:
 When a Pizza is cut into eight pieces as shown, each of the angles1 to 8 formed at the center are acute angles.

### Solved Example

Question: In the adjoining diagram QT is perpendicular to PR. Identify the acute angles so formed.

Solution:

Angles PQT and RQT are right angles
Hence, ∠PQU,   ∠UQT,  ∠TQS and  ∠SQR are all acute angles.

We cannot classify ∠UQS as we do not know the measures of angles UQT and TQS.

## Acute Angle in Real Life

We come across acute angles in many a real life situation. Let us see few examples for this.

In the example below a cross road is shown making an acute angle with the main road on one side.

• When you place a ladder against the wall, it makes two angles as shown in the sketch, one angle with the floor and one with the wall. You can notice that both of them are acute angles.

• In a similar manner, a flight of stairs also make acute angles with the floors it connects.
• A sliding ramp kept for children in a park also makes an acute angle with the ground.

The shorter side of the famous leaning tower of Pisa makes an acute angle with the level ground.
• When a car hood is kept open to inspect the engine, you can observe the acute angle made.
A not fully open kept door makes an acute angle with the vertical frame to which it is fixed. Similarly if a window is kept half open, an acute angle is formed between the frame of the window and its closing door.