Acute Angle - Definition with Examples

What is an Acute Angle?

An acute angle is an angle whose measure is greater than 0 degrees and less than 90 degrees. In other words, an acute angle is an angle that is smaller than a right angle.

This type of angle is an acute angle

The other side’s large angle is called Reflex Angle.

Acute angle Images

Look below Angles’ images. These all are acute angles images:

Definition of Acute Angle

An acute angle, in geometry, refers to an angle that measures less than 90 degrees. It is characterized by its openness, as it forms a sharp corner or point between two intersecting lines or line segments.

In simpler terms, an acute angle is narrower than a right angle (90 degrees) but wider than a straight angle (180 degrees). It can be visualized as the angle formed when you fold the corner of a piece of paper or when two hands on a clock point to numbers less than 3 and 9 on the clock face. The term “acute” comes from the Latin word “acutus,” meaning “sharp” or “pointed,” emphasizing the sharpness of the angle.

Acute Angle Degree

An acute angle, in terms of degrees, is an angle that measures less than 90 degrees but greater than 0 degrees. Specifically, it falls within the range of 0 degrees to 89 degrees. The acute angle is characterized by its sharpness and openness, forming a narrow corner or point where two lines or line segments intersect.

For example, an acute angle could measure 30 degrees, 45 degrees, or any value between 0 and 89 degrees. The degree measurement of an acute angle helps us understand its magnitude and relationship to other angles within a geometric context.

Real-life Examples of Acute Angles

1. pizza slices

If you slice a pizza 5,6 or more slices, then each slice of pizza will make an acute angle.

2. Wall clock

The wall clock is another example of an acute angle. It makes acute angles at numerous hours of the day. For example, 10 o’clock.

Acute Angle Triangle

An acute triangle is a type of triangle in which all three angles are acute angles, meaning each angle measures less than 90 degrees. It is characterized by its sharp corners or acute angles. In an acute triangle, the sum of the measures of all three angles is always less than 180 degrees.

Properties of an Acute Triangle

Angle Measurements: All three angles in an acute triangle are less than 90 degrees. This means that no angle within the triangle is a right angle (90 degrees) or an obtuse angle (greater than 90 degrees).
Side Lengths: In an acute triangle, the lengths of the sides can vary. There are no specific restrictions on the lengths of the sides.
Perpendicular Bisectors: The perpendicular bisectors of the sides of an acute triangle intersect inside the triangle, forming a point called the circumcenter. The circumcenter is equidistant from the three vertices of the triangle.
Altitudes: The altitudes of an acute triangle, which are the perpendiculars drawn from each vertex to the opposite side, always lie inside the triangle.
Area: The area of an acute triangle can be calculated using the formula: Area = (1/2) * base * height, where the base is any side of the triangle, and the height is the perpendicular distance from the base to the opposite vertex.

Acute Angle Formula

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:In a triangle, the measures of the three angles are 40 degrees, 60 degrees, and 80 degrees. Determine if it is an acute triangle.

To determine if the triangle is acute, we need to verify that all three angles are less than 90 degrees. Given that the measures of the angles are 40 degrees, 60 degrees, and 80 degrees, we can see that all angles are less than 90 degrees. Therefore, this triangle is an acute triangle.

Example 2:Given a triangle with side lengths of 5 cm, 7 cm, and 9 cm, determine if it is an acute triangle.

To determine if the triangle is acute, we can use the triangle inequality theorem, which states that the sum of the lengths of any two sides of a triangle must be greater than the length of the third side.

Let’s check the inequalities:
5² + 7² > 9² (25 + 49 > 81) – True
7² + 9² > 5² (49 + 81 > 25) – True
9² + 5² > 7² (81 + 25 > 49) – True

Therefore, this triangle is an acute triangle.

FAQs

Q: What is an acute angle?

A: An acute angle is an angle that measures less than 90 degrees. It is characterized by its sharpness and openness, forming a narrow corner or point where two lines or line segments intersect.

Q: Can an acute triangle have equal side lengths?

A: Yes, an acute triangle can have equal side lengths. It is possible for all three sides of an acute triangle to be of equal length, forming an equilateral acute triangle. In this case, all three angles of the triangle would measure 60 degrees.

Q: What are some properties of acute triangles?

A: Here are some properties of acute triangles:

All three angles are acute (less than 90 degrees).
The sum of the measures of the angles is always less than 180 degrees.
The perpendicular bisectors of the sides intersect inside the triangle.
The altitudes of the triangle (perpendiculars drawn from each vertex to the opposite side) lie inside the triangle.
The area of an acute triangle can be calculated using the formula: Area = (1/2) * base * height, where the base is any side of the triangle and the height is the perpendicular distance from the base to the opposite vertex.

Q: What are some real-life examples of acute triangles?

A: Acute triangles can be found in various real-life situations, including:

Roof structures: The shape of many roofs, such as gable roofs, often involves acute triangles.
Road signs: Triangular road signs, such as yield signs or warning signs, often have acute angles.
Pyramids: The faces of pyramids are triangular, and the angles of these faces are acute.
Mountain peaks: The shape of many mountain peaks forms acute triangles.

Acute Angle – Definition with Examples