Obtuse Angle
In geometry, an obtuse angle is defined as an angle that is greater than 90° and less than 180°. Often during a day in a 24 hours duration, we can see a clock framing many obtuse angle degrees between a minute hand and an hour hand. Let us learn more about the obtuse angle and its properties.
| 1. | What is an Obtuse Angle? |
| 2. | Obtuse Angle Degree |
| 3. | Obtuse Angle in Real Life |
| 4. | Obtuse Angle Triangle |
| 5. | Acute and Obtuse Angles |
| 6. | FAQs on Obtuse Angle |
What is an Obtuse Angle?
The definition of an obtuse angle in geometry states that 'an angle whose measure is greater than 90° and less than 180° is called an obtuse angle'. In other words, an obtuse angle is an angle between a right angle and a straight angle. Look at some of the examples of obtuse angles given below.
Obtuse Angle Degree
In the above section, we read that an angle that measures less than 180 degrees and more than 90 degrees angle is called an obtuse angle. The examples of obtuse angle degrees are 165°, 135°, 110°, 179°, 91°, etc. Hence, the obtuse angle degree lies within the ranges from 90° to 180°.
Obtuse Angles in Real Life
We know that angles measuring greater than 90° and less than 180° are called obtuse angles. Here are some real-life examples of obtuse angles. Can you observe the obtuse angles in all these images? Can you think of more objects in real life that include obtuse angles?
Some other examples of obtuse angles in real-life are given below:
- The angle between the hour and minute hand of a clock at 4 o'clock.
- The angle between the base of an open laptop and its screen.
- Angles formed by the blades of a ceiling fan.
Obtuse Angle Triangle
When one of the vertex angles of a triangle is greater than 90°, it is called an obtuse angle triangle. An obtuse triangle can either be an isosceles or a scalene triangle. An equilateral triangle cannot be obtuse. The side opposite to the obtuse angle in the triangle is the longest side of that triangle. Similarly, a triangle can never be a right angle and an obtuse angle at the same time as per the angle sum property of a triangle. Thus, we can conclude that if one of the angles of a triangle is obtuse, then the other two angles of a triangle must be acute angles.
The triangles above have one angle greater than 90°. Hence, they are called obtuse-angled triangles or simply obtuse triangles. In an obtuse angle triangle, the sum of the squares of the two sides is less than the square of the longest side. In ΔABC, the sides measure a,b,c such that c is the largest side, we have: a2 + b2 < c2. Conversely, if in a triangle, if a2 + b2 < c2, then the triangle is an obtuse triangle.
Acute and Obtuse Angles
There are different types of angles based on the measurement. Angles that measure less than 90 degrees are known as acute angles, while the angles greater than 90° but less than 180° are known as obtuse angles. Look at the image of acute and obtuse angles given below followed by their difference.
Now, let us understand acute vs obtuse angle through the table below:
| Acute Angle | Obtuse Angle |
|---|---|
| It measures less than 90 degrees. | It measures greater than 90 and less than 180 degrees. |
| A triangle with three acute angles is known as an acute angle triangle. | A triangle with 1 obtuse angle and 2 acute angles is termed an obtuse angle triangle. |
| Examples of acute angles: 56°, 12°, 79°, 43°, etc. | Examples of obtuse angles: 124°, 179°, 150°, 95°, etc. |
Important Notes
- All angles measuring more than 90° and less than 180° are called obtuse angles.
- In an obtuse angle triangle, the sum of the squares of the two sides is less than the square of the longest side.
Thinking Out Of the Box!
- Can a triangle have more than one obtuse angle?
- Which polygon has all its internal angles as obtuse angles?
- Can there be a parallelogram without an obtuse angle? If so, what shape can it be?
Topics Related to Obtuse Angles
Check out a few more interesting articles related to the obtuse angle and its properties.
Obtuse Angle Examples
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Example 1: Identify the obtuse angles from the following figures.
Solution:
Option (b) - 117° and option (c) - 121° are more than 90° and less than 180°. Therefore, they are obtuse angles. Therefore, 117° and 121° are obtuse angles.
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Example 2: At what times, in the clocks shown below, an obtuse angle is formed?
Solution:
We can observe that in all instances, an obtuse angle is formed between the hour's hand and the minutes' hand of the clock.
- At 5:00, the hour hand is at 5 and the minute hand is at 12. So, by observation, we can conclude that the angle formed is greater than 90º but less than 180°, which is an obtuse angle. Using a protractor, we can confirm that it is making an angle between (90° to 180°).
- At 8:00, the hour hand is at 8 and the minute hand is at 12. So, by observation, we can conclude that the angle formed is greater than 90º but less than 180°. Thus it is an obtuse angle.
- At 10:15, the hour hand is at 10 and the minute hand is at 3. So, by observation, we can conclude that the angle formed is an obtuse angle.
- At 2:40, the hour hand is slightly above 3 and the minute hand is at 8. So, by observation, we can conclude that the angle formed is greater than 90º but less than 180º. Using a protractor, we can confirm that it is making an obtuse angle.
Therefore, an obtuse angle is formed at 5:00, 8:00, 10:15, and 2:40.
FAQs on Obtuse Angle
What is an Obtuse Angle in Geometry?
In geometry, we read about various types of angles. Each type of angle has its own characteristics with the help of which we can identify it. Similarly, an obtuse angle is an angle that is always less than 180 degrees and is greater than 90 degrees.
What does an Obtuse Angle Look Like?
The angles measuring greater than 90° and less than 180° are called obtuse angles in geometry. The obtuse angle lies between 90° and 180° and looks like a reclined chair, an angle below the staircase, or an angle formed between a minute and an hour hand of a clock at 10:15 AM.
How do you Draw an Obtuse Angle?
The definition of an obtuse angle in geometry states that an angle larger than 90° but less than 180° is called an obtuse angle. We can use a protractor and mark any angle between 90° and 180° to make an obtuse angle.
What are Some Examples of Obtuse Angles?
145°, 150°, 178°, 149°, 91° are all examples of obtuse angle degrees as they are more than 90° and less than 180°.
What are the Properties of Obtuse Angles?
Given below are the properties of an obtuse angle:
- Angles that measure between 90 and 180 degrees.
- Angles lie between a right and a straight angle.
Can Two Obtuse Angles be Supplementary?
Two obtuse angles, each measuring greater than 90° cannot form a supplementary pair of angles as the sum will be greater than 180° which doesn't satisfy the condition of supplementary angles.
What is Obtuse Angle Triangle?
A triangle with 1 obtuse angle and the other two acute angles is an obtuse-angled triangle. The sum of all three interior angles will be 180 degrees.
What is Acute and Obtuse Angle?
An acute angle is the one that measures less than 90 degrees and the angles greater than 90° but less than 180° are obtuse angles.
How many Degrees is an Obtuse Angle?
The degree of an obtuse angle always lies between 90° to 180°.