Acute Angle

An angle smaller than the right angle is called an acute angle. In other words, the angle which is less than 90 degrees forms an acute angle. The polygons such as triangle, parallelogram, trapezoid, etc. consist of at least one acute angle in it. Let us learn here some basics of angles.

The two lines or line segments or rays with a common point leads to the formation of an angle. An angle is denoted through a symbol ‘∠’. An angle is measured with the help of a protractor in degrees (o).

  • Vertex: A corner or a point where lines meet is called the vertex of an angle.
  • Arms: The two lines that meet to make an angle are called arms of an angle.

Acute Angle

  • Measuring Angle: Angles can be measured using a protractor.
  1. Place the center of the protractor at a fixed point, say O of the angle at one of the arms.
  2. Another arm indicates the degree of the angle while measured from the initial hand in an anti-clockwise manner.

Acute Angle Definition

An angle which is measuring less than 90 degrees is called an acute angle. This angle is smaller than the right angle (which is equal to 90 degrees). For example, ∠30o, ∠45o, ∠60o, ∠75o, ∠33o, ∠55o, ∠85o, etc. are all acute angles. Let us see some examples here to understand.

Example: See the figure below:

Acute angle definition

∠ABC measures 30 degrees, which is less than 90 degrees and hence it is an acute angle.

The basic types of angles are right angle, obtuse angle, acute angle.

Types of Angles

From the above figure, we can differentiate between the three angles. The first figure shows an acute angle measuring 62 degrees (less than 90 degrees), the second figure shows right angle measuring 90 degrees and the third figure shows obtuse angle measuring 135 degrees (more than 90 degrees).

Note: Dividing the right angle gives us two or more acute angles since each angle formed, there will be much less than 90˚.

Acute angle formation

Also, read:

Acute Angle Triangle

If all the internal angles of a triangle are less than 90 degrees is called an acute angle triangle.

Acute angle triangle

In the above figure, we can see, the three angles of the triangle are 69, 85 and 26. All three angles are less than 90 degrees.

Properties of Acute Triangles

  • All equilateral triangles are acute triangles. An equilateral triangle has three sides of the same length and three angles of the equal measure, i.e. 60°
  • Opposite the highest angle is the longest side of an acute triangle
  • Acute triangles can be isosceles, equilateral, or scalene

Acute Angle Parallelogram

A parallelogram is a quadrilateral with opposite sides being parallel and congruent (of the same length). The angles that are opposite to each other are congruent (of equal measure). A parallelogram has two acute angles and two obtuse angles. Therefore, the acute and obtuse angles should have the equal measurement.

acute angle of a parallelogram

Acute Angle Trapezoid

A trapezoid is a parallelogram with one pair of opposite sides being parallel. An acute trapezoid has both interior angles (created by the base and legs which are longer) measuring less than 90°.

Example: There are two acute and two obtuse angles in the trapezoid given below.

Acute angle trapezoid

Acute Angle Formula

In an acute triangle, the following statement holds good for the length of the sides:

a2 + b2 > c2

b2 + c2 > a2

c2 + a2 > b2

Where a, b and c are the sides of a triangle.

Acute Angle Examples

Example 1:

Angle A measures x degrees. Is A acute if x = 15? If x = 65? If x = 90? If x = 135?

15 < 90 ? Yes. ∠A is acute if x = 15.

65 < 90 ? Yes. ∠A is acute if x = 65.

90 < 90 ? No. ∠A isn’t acute if x = 90.

135 < 90 ? No. ∠A isn’t acute if x = 135.

Example 2:

Which angle is right? Acute? Obtuse?

Acute angle example

Answer: Angle A is acute, angle B is right, and angle C is obtuse.

Acute Angles in Real Life

  • The angle made in letter V is an acute angle.
  • If we slice a pizza into four or more slices, each slice of pizza will make an acute angle.
  • The hands of a wall clock make acute angles at several hours of a day. Example, at 1 o’clock.
  • The road signs, namely “One Way” and “No Left Turn” arrows show an acute angle.
  • The turn signal indicator of the dashboard and the speedometer both establish acute angles inside the vehicle.

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