An acute angle is an angle that measures less than 90 degrees but more than 0 degrees. These angles are smaller than a right angle and are often found in triangles and various geometric shapes. Imagine the angle formed when you bend your elbow, creating a sharp but not wide opening; this is an example of an Acute Angle. These angles play a significant role in diverse mathematical and scientific scenarios.
In this article, we have covered the various concepts related to acute angles—definition, properties, and real-life examples of acute angles to gain a clearer understanding of their significance.
Acute Angle
An acute angle is a small angle that measures less than 90° and greater than 0° .
Shape of Acute Angle
The acute angle looks like a small wedge or a slice of pizza. It is less than 90°, and it points in a narrow way. Imagine a corner where two walls meet, and the angle between them is sharp and small. That's what an acute angle is like—pointed and less than a right angle.
- The vertex is the point where the two arms of the angle meet. It's like the tip of the slice in the pizza analogy.
- The arms of the angle are the two lines that come out from the vertex, like the two sides of the pizza slice. They spread apart from each other but don't go too far, staying less than a right angle (90°).
Properties of Acute Angle
The properties of the acute angle are:
- An acute angle has a measurement greater than zero, denoted as 0° < θ.
- The measurement of an acute angle is less than that of a right angle, specifically 90° > θ.
- The sine (sin) of any acute angle falls between 0 and 1.
- The cosine (cos) of any acute angle falls between 1 and 0.
- The tangent (tan) of any acute angle falls between 0 and infinity (∞).
Acute Angle Degree
The degree of an acute angle, which is any angle smaller than 90°, can be represented by the formula:
0° < Measure of Acute Angle < 90°
This means that the measurement of an acute angle falls within the range from 0° to less than 90°. Examples of acute angle° include 63°, 31°, 44°, 68°, 83°, and 85°. The formula provides a guideline for understanding and identifying acute angles based on their degree measurements.
Acute Angle Triangle
An Acute Angled Triangle possesses angles that are all smaller than 90°. When all three angles in a triangle measure 60°, it becomes a special type known as an equilateral triangle.
Acute triangles can be categorized into acute scalene triangles, acute isosceles triangles, and equilateral triangles. The term "acute triangle" refers to a type of triangle where all interior angles are less than 90°. In the illustrated triangle below, all its angles are smaller than 90°, leading to its classification as an acute triangle.
An acute
In an acute angle triangle, there's a rule called the triangle inequality theorem. It's like a formula which saying that if a triangle has sides named a, b, and c (where c is the longest side), then a squared plus b squared is more than c squared. So, if a squared plus b squared is greater than c squared, you have an acute triangle. This rule helps us understand the relationships between the sides of a triangle when it's an acute one.
- a2 + b2 > c2
- c2 + b2 > a2
- a2 + c2 > b2
Acute Angle in Various Shapes
Acute angles can be found in various geometric shapes. Here are some shapes that often contain acute angles:
- Triangle
- Quadrilateral
- Pentagon
- Polygons
Let's discuss acute angles in these shapes in detail.
Acute Angle in Triangle
In a triangle, an acute angle is an angle that measures less than 90°. Imagine a triangle with one corner pointing outward but not too widely. For example, in a right-angled triangle, the angle opposite the right angle is acute. If the angle is less than 90°, it's an acute angle in a triangle.
Acute Angle in Quadrilateral
In a quadrilateral, which is a four-sided shape, an acute angle is an angle measuring less than 90°. Unlike right angles found in squares or rectangles, acute angles in a quadrilateral are smaller, creating a sharper corner. For example, consider a kite-shaped quadrilateral where the angles formed are acute.
Acute Angle in Polygon
In a regular polygon, where all sides and angles are equal, acute angles are not present. Consider a regular pentagon, hexagon, or any other polygon with equal sides and angles. In these shapes, all interior angles are equal and measure more than 90°, making them obtuse. Acute angles are typically found in irregular polygons, where sides and angles can vary, allowing for angles measuring less than 90°.
Acute Angle in Irregular Shapes
Irregular shapes can also have acute angles. These are angles that are less than 90° but might be found in shapes that don't follow a regular pattern. For example, in a shape with uneven sides and angles, any angle measuring less than 90° is an acute angle.
Acute Angle in Composite Shapes
Composite shapes, formed by combining two or more simple shapes, can contain acute angles. Picture a shape made by joining a rectangle and a triangle. The angle where the two shapes meet can be acute if it measures less than 90°. In composite shapes, any angle smaller than 90° is an acute angle.
Real Life Example of Acute Angle
Some examples of Real-life scenarios which looks like Acute angle are:
- Clock Hands: When the minute hand on a clock points at 3 and the hour hand is at 4, the angle between them forms an acute angle. It's less than 90°.
- Roof Design: In the roof of a house, the intersection of two inclined sides often forms acute angles, especially at the peak of the roof.
- Triangle in a Playground Slide: The shape of a playground slide often involves acute angles, particularly at the top where the two sides of the slide come together.
- Pizza Slices: When you cut a pizza into slices, each slice is like a triangle with acute angles, meeting at the center of the pizza.
- Road Signs: The shape of some road signs, like yield signs or diamond-shaped warning signs, may have acute angles formed by their edges.
Acute Angle, Obtuse Angle and Right Angle
- Acute Angle: An acute angle is an angle whose measure is less than 90°.
- In an acute-angled triangle, all three angles are acute angles. An example of an acute angle is an angle measuring 60°.
- Obtuse Angle: An obtuse angle is an angle whose measure is greater than 90° but less than 180°.
- In an obtuse-angled triangle, one of the angles is an obtuse angle. An example of an obtuse angle is an angle measuring 120°.
- Right Angle: A right angle is an angle whose measure is exactly 90°.
- In a right-angled triangle, one of the angles is a right angle.
Difference Between Acute and Obtuse Angle
Key difference between acute and obtuse angles are listed in the following table:
Acute vs Obtuse Angle |
|---|
Characteristics | Acute Angle | Obtuse Angle |
|---|
Definition | An angle whose measure is less than 90°. | An angle whose measure is greater than 90° but less than 180°. |
|---|
Angle Measure | Less than 90° | Between 90° and 180° |
|---|
Triangle Type | Can be found in an acute-angled triangle | Can be found in an obtuse-angled triangle |
|---|
Appearance | Often appears sharper or narrower | Often appears wider or more open |
|---|
Relationship with right angle | Smaller than a right angle | Larger than a right angle |
|---|
Example | An angle measuring 60°. | An angle measuring 120°. |
|---|
Symbol | ∠ABC, where m∠ABC < 90° | ∠PQR, where 180°>∠PQR > 90° |
|---|
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Solved Examples on Acute Angle
Example 1: Which of the following angles given in the figure 1 are acute angles? Give reason for why and why not.
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Solution:
a. 45°: Yes, it is an acute angle. An acute angle is any angle smaller than 90°, and 45° falls within this range.
b. 90°: No, it is not an acute angle. An acute angle must be less than 90°, and a 90° angle is called a right angle.
c. 120°: No, it is not an acute angle. Acute angles are smaller than 90°, and 120° exceeds this limit.
d. 60°: Yes, it is an acute angle. A 60° angle is smaller than 90°, fitting the criteria for an acute angle.
Example 2: What is the sum of the interior angles in an acute triangle if one angle measures 30°, another angle measures 45°, and the third angle is 60°?
Solution:
To find the sum of the interior angles in the given acute triangle, add the measures of the three angles:
30°+45°+60°=135°.
Therefore, the sum of the interior angles in the acute triangle is 135°
Practice Questions on Acute Angle
Question 1: Determine if the following sets of angles form an acute triangle.
- 30°, 50°, 80°
- 60°, 70°, 80°
- 45°, 45°, 90°
- 100°, 40°, 40°
Question 2: Find the sum of the acute angles in the triangles.
- Triangle ABC has angles 30°, 45°, 60°.
- Triangle XYZ has angles 20°, 70°, 80°.
- Triangle PQR has angles 50°, 30°, 60°.
Question 3: If an angle is 25°, what is its complement, and is it an acute angle?
Question 4: Classify each triangle based on its angles.
- Triangle LMN with angles 60°, 70°, 50°.
- Triangle DEF with angles 80°, 60°, 40°.
- Triangle UVW with angles 30°, 60°, 90°.
Conclusion
Acute angles are important concept in geometry, as they are angles that measure less than 90 degrees. They can be found in many shapes, patterns, and real-world objects, making them essential to understanding basic math concepts. By recognizing and working with acute angles, students can strengthen their geometry skills and gain a deeper understanding of how angles help us describe and measure the world around us.