An obtuse angle is a fundamental concept in geometry that describes an angle larger than a right angle, measuring between 90 and 180 degrees. Unlike acute angles, which are smaller than 90 degrees and right angles, which measure exactly 90 degrees, an obtuse angle exhibits an opening wider than a right angle. Visualizing an obtuse angle, it appears wider than a simple corner, forming a broad “V” or “U” shape. Understanding and identifying obtuse angles are essential in geometry, aiding in various geometric constructions, measurements, and spatial problem-solving.
What is an Obtuse Angle?
An obtuse angle is a geometric term used to describe an angle that measures more than 90 degrees but less than 180 degrees. It appears wider or more open than a right angle but narrower than a straight angle. This angle is commonly observed in various geometric shapes and can be recognized by its measure falling between a right angle and a straight angle.
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Obtuse Angle Definition
The angle which measure greater than 90 degrees and less than 180 degrees is called Obtuse Angle
Obtuse Angle Example
An obtuse angle, typically found in triangles, refers to an angle measuring more than 90 degrees but less than 180 degrees. This specific type of angle deviates from a right angle (90 degrees) by being wider but doesn’t reach the linearity of a straight angle (180 degrees). For instance, if one angle within a triangle measures 120 degrees it indicates an angle that surpasses the 90-degree threshold yet doesn’t extend to the full 180 degrees. This scenario illustrates an obtuse angle within the triangle, representing an angle greater than 90 degrees but less than 180 degrees.
Obtuse Angle Formula
An obtuse angle symbolized by spans beyond a right angle (measuring 90 degrees) but falls short of forming a straight line (measuring 180 degrees). It occupies a range greater than 90 degrees yet less than 180 degrees on the standard angle scale. Visually, it’s wider than a right angle creating a broader gap between its rays without stretching into a complete straight line. Essentially, it’s an angle between 90 and 180 degrees distinct from acute angles and straight angles.
90 degrees < θ < 180 degrees
Obtuse Angle Real Life Examples
There are many real-life examples of obtuse angles. Some of them are listed below:
- The angle formed by the minute and hour hands on a clock when it is between 3 and 6.
- Angle between laptop and screen.
- Angle of reclining chair.
Obtuse Angle Shape
The shapes which contain at least one obtuse angle falls under the obtuse angle shape. Some of the obtuse angle shapes are obtuse angled triangle, parallelogram, rhombus and many more.
Obtuse Angle Degree
An obtuse angle is a type of angle that is more than a right angle (which measures exactly 90 degrees) but not as wide as a straight angle (which measures 180 degrees). Visually, an obtuse angle forms a wider opening than a right angle but doesn’t form a full straight line like a straight angle. It typically falls between 90 and 180 degrees.
Obtuse Angle Triangle
A Triangle whose one of the angle is greater than 90 degree but less than 180 degrees is called Obtuse Angled Triangle. In a triangle we can have only one obtuse angle. A triangle with one of its interior angles is known as the obtuse angled triangle. An obtuse angled triangle has one vertex with obtuse angle and other two vertices with acute angles. In this triangle the side opposite to the obtuse angle is the longest side of the triangle.
Obtuse Angle Parallelogram
An obtuse angle within a parallelogram refers to an angle within the shape that measures more than 90 degrees but less than 180 degrees. In a parallelogram, the opposite angles are equal, which means that if one angle is obtuse, its opposite angle across the parallelogram will also be obtuse and congruent in measurement. This characteristic is consistent with the properties of parallelograms where opposite angles are equal and the sum of adjacent angles is always 180 degrees.
Right, Acute, Obtuse Angle
The angle less than 90 degree is called acute angle. The angle equals to 90 degree is called right angle. The angle greater than 90 degree and less than 180 is called obtuse angle.
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The angle that is equal to 90 degrees is called right angle.
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The angle that is less than 90 degrees is called acute angle.
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The angle that is greater than 90 degrees is called acute angle.
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.png) Right Angle |
.png) Acute Angle |
.png) Obtuse Angle |
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Solved Examples on Obtuse Angle
Example 1: In triangle PQR, if angle P measures 60 degrees and angle Q measures 45 degrees, determine the measure of angle R.
Solution:
For triangle PQR, the sum of its interior angles equals 180 degrees. Using this:
Angle P + Angle Q + Angle R = 180 degrees
Given that Angle P = 60 degrees and Angle Q = 45 degrees:
60 degrees + 45 degrees + Angle R = 180 degrees
105 degrees + Angle R = 180 degrees
Angle R = 180 degrees – 105 degrees
Angle R = 75 degrees
Example 2: The exterior angle of a polygon is measured at 130 degrees. Calculate the corresponding interior angle.
Solution:
For any polygon, the exterior angle plus its corresponding interior angle forms a straight line, which measures 180 degrees. Therefore:
Exterior angle + Interior angle = 180 degrees
Given the exterior angle = 130 degrees:
130 degrees + Interior angle = 180 degrees
Interior angle = 180 degrees – 130 degrees
Interior angle = 50 degrees
Example 3: If an angle measures 160 degrees, find its complement.
Solution:
The complement of an angle is the difference between the angle and 90 degrees.
Given angle = 160 degrees:
Complement = 90 degrees – 160 degrees
Complement = -70 degrees
However, as a complement can’t be negative, it implies that the given angle of 160 degrees is an obtuse angle itself and doesn’t have a complement within the defined range of angles (0 to 90 degrees).
Example 4: In a quadrilateral, one of the angles measures 125 degrees. Is this angle obtuse?
Solution:
Yes, an angle measuring 125 degrees is greater than 90 degrees, so it is classified as an obtuse angle.
Example 5: An angle measures 150 degrees. If it is a part of a triangle can this angle be an obtuse angle?
Solution:
Absolutely, since an obtuse angle falls between 90 and 180 degrees an angle measuring 150 degrees fits into the range and can be classified as obtuse.
Example 6: In a pentagon one angle measures 110 degrees and another measures 95 degrees. Are these angles obtuse?
Solution:
Among the two angles given the 110-degree angle is an obtuse angle as it is larger than 90 degrees whereas the 95-degree angle is acute as it’s less than 90 degrees.
Example 7: Within a hexagon, an angle measures 135 degrees. Does this angle fit the description of an obtuse angle?
Solution:
Yes, an angle that measures 135 degrees surpasses 90 degrees yet remains less than 180 degrees, aligning with the characteristics of an obtuse angle.
Example 8: If an angle measures 170 degrees, would it qualify as an obtuse angle?
Solution:
Absolutely, an angle that measures 170 degrees falls within the range of obtuse angles since it exceeds 90 degrees but remains below 180 degrees. Hence, it is categorized as an obtuse angle.
Obtuse Angle – Practice Questions
Q1: Identify an obtuse angle in your surroundings and measure its approximate degree.
Q2: Given an angle measuring 100 degrees, determine if it is acute, obtuse, or neither.
Q3: Draw an obtuse angle using a protractor and measure its degree.
Q4: Calculate the measure of an obtuse angle that is supplementary to a 60-degree angle.
Q5: Identify and classify three obtuse angles within a shape or figure.
Obtuse Angle – FAQs
1. What is the Opposite of an Obtuse Angle?
The opposite of an obtuse angle is an acute angle, which measures less than 90 degrees.
2. Can an Obtuse Angle be a Part of a Right-Angled Triangle?
No, by definition, an obtuse angle is larger than a right angle, which measures 90 degrees. Therefore, it cannot be part of a right-angled triangle.
3. Are there Real-Life Examples of Obtuse Angles?
Yes, examples include the angle formed by the minute and hour hands on a clock when it is between 3 and 6.
4. How does Understanding Obtuse Angles Benefit in Real-World Applications?
Understanding obtuse angles is crucial in fields like architecture and engineering, where precise angle measurements are essential for designing structures and systems.
5. Can an Obtuse Angle be Complementary to Another Angle?
No, because the sum of two complementary angles is always 90 degrees, and an obtuse angle already exceeds this measure.
6. What is Obtuse Angle Triangle?
A triangle with an obtuse angle is called as obtuse angle triangle.
7. Can Two Obtuse Angle be Supplementary?
No, two obtuse angles cannot be supplementary as the sum of two obtuse angle is greater than 180 degrees.
8. How Many Degrees is an Obtuse Angle?
Degrees greater than 90 degree and less than 180 degree is an obtuse angle.
9. What is Acute and Obtuse Angle?
The angle less than 90 degree is called acute angle and the angle greater than 90 degree and less than 180 degrees is called obtuse angle.
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