Obtuse Angle
There are different types of angles formed on the plane surface and an obtuse angle is one of those angles. In Geometry, an obtuse angle is an angle that is greater than 90° and less than 180°.
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.
Examples of Obtuse Angles
We know that angles measuring greater than 90° and less than 180° are called obtuse angles. Therefore, angles that measure 145°,150°,178°,149°, 91° are considered as obtuse angle examples. 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?
Obtuse Angle of a Triangle
When one of the vertex angles of a triangle is greater than 90°, it is called an obtuse triangle. An obtuse triangle can either be an isosceles or a scalene triangle. An equilateral triangle cannot be obtuse. The angle opposite to the obtuse angle in the triangle is the longest side of that triangle. Similarly, a triangle can never 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.
The triangles above have one angle greater than 90°. Hence, they are called obtuse-angled triangles or simply obtuse triangles. In an obtuse 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, thus: a2 + b2 < c2. Conversely, if in a triangle, if a2 + b2 < c2, then the triangle is an obtuse triangle.
Obtuse Angles of a Rhombus
A rhombus is a special type of quadrilateral which includes:
- four equal sides
- two pairs of parallel sides
- equal opposite angles
The sum of the interior angles of any quadrilateral is 180° and in a rhombus, consecutive angles are supplementary and opposite angles are equal. Thus, at any given time, a rhombus has two obtuse angles that are equal and the other two angles are acute and they are also equal.
Obtuse Angles of a Parallelogram
A Parallelogram is a special type of quadrilateral which includes:
- two pairs of parallel sides
- opposite sides of equal lengths
- equal opposite angles
The sum of the interior angles of any quadrilateral is 180° and in a parallelogram, consecutive angles are supplementary and opposite angles are equal. Thus, at any given time, a parallelogram has two obtuse angles that are equal and the other two angles are acute and they are also equal.
Important Notes
- All angles measuring more than 90° and less than 180° are called obtuse angles.
- In an obtuse 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
Solved Examples on Obtuse Angle
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Example 1: Identify the obtuse angles from the following figures.
Solution:
Option (b) and option (c) are more than 90° and less than 180°. Therefore, they are obtuse angles. Option (b) and option (c) 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 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 180º, which is an obtuse angle. Using a protractor, we can confirm that since it is making an angle of 150º, thus it is an obtuse angle only.
- 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 180º, which is an obtuse angle. Using a protractor, we can confirm that since it is making an angle of 240º, thus it is an obtuse angle only.
- 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 greater than 90º but less than 180º, which is an obtuse angle. Using a protractor, we can confirm that since it is making an angle of 150º, thus it is an obtuse angle only.
- At 2:40, the hour hand is slightly above and the minute hand is at 8. So, by observation, we can conclude that the angle formed greater than 90º but less than 180º, which is an obtuse angle. Using a protractor, we can confirm that since it is making an angle of 150º, thus it is an obtuse angle only.
Therefore, an obtuse angle is formed at 5:00, 8:00, 10:15, and 2:40
FAQs on Obtuse Angle
How do you Create 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 angles as they are more than 90° and less than 180°.
How do you Determine an Obtuse Angle?
The definition of an obtuse angle states that if an angle measures more than 90° and less than 180°, it is an obtuse angle. We can use a protractor and verify the same.
What are the Properties of the Obtuse-angled Triangle?
Given below are the properties of an Obtuse Angled Triangle Properties
- In a triangle, the sum of the two angles except the obtuse angle is less than 90 degrees.
- The side opposite to the obtuse angle in a triangle is the longest side of the triangle.
- An obtuse triangle can have one and only one obtuse angle to satisfy the angle sum property of the triangle.
Can 2 Obtuse Angles be Supplementary?
Two obtuse angles, each measuring greater than 90° cannot form a supplementary pair of angles as then the sum will be greater than 180° which doesn't satisfy the definition of supplementary angles.