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# Obtuse Angle|Definition & Meaning

**Definition**

In geometry, an obtuse angle is one with a value **larger** than **90 degrees** but less than **180 degrees**. The phrase “an angle with measurement exceeding 90 degrees but less than 180 degrees is termed an **obtuse angle**.” To put it another way, an obtuse angle is one that falls in the middle of a right angle and a 90-degree angle.

One of the triangle’s angles is obtuse. Hence, this type of triangle is called an **obtuse triangle**. There is no such thing as a triangle that is composed entirely of obtuse angles because that would not even qualify as a shape called a triangle. Obtuseness is a property that can only be shared by only one angle in a triangle.

## Degree of Obtuse Angle

According to what we studied in the section that came before this one, an **obtuse angle** is one that has a measurement that is **fewer** than **180 degrees **but **greater **than **90 degrees**. **165** degrees, **135 **degrees, **110 **degrees, **179** degrees,** 91** degrees, and so on are all examples of obtuse angles. As a result, the degree of obtuse angle falls somewhere between the ranges of **90** and **180** degrees.

These angles have a measurement that falls between 90 and 180 degrees.

A right angle, which measures **90** degrees, is greater than an acute angle, whose measurements is fewer than **90** degrees, but an obtuse angle, which measures less than **180 **degrees, is less than a straight line angle which is **180** degree. On the other hand, if we list the angles from smallest to largest measurement, the order would be as follows: **acute**, **right**, **obtuse**, and **straight**.

This angle represents **greater** than a **circle quarter** yet **fewer** than a **circle half **when viewed visually. Therefore, if we cut a circle into quarters, an obtuse angle always will fill between one-fourth and one-half of the circle when we do this.

Within the confines of an obtuse triangle, there is invariably an angle that is obtuse.

## Obtuse Angle In Triangle

A triangle with an obtuse angle is a type of triangle that is defined by having **one** of its vertex angles that is more than **90** degrees. You can have an **isosceles **triangle or a **scalene **triangle when you have an** obtuse** triangle. It is impossible for a triangle with equal sides to have an obtuse angle.

The side of the triangle that is perpendicular to an obtuse angle is the largest of the three of that particular triangle.

According to the **property** of the **sum** of the angles of the triangle, there is **no** way for there to be both an **obtuse **angle and a **right** angle within the same triangle at the same time.

Therefore, we are able to draw the conclusion that if one of the angles that make up a triangle is an **obtuse** angle, then the remaining two angles that make up a triangle should be acute angles.

One of the triangles up top has an angle that is larger than **ninety** degrees. Because of this, we refer to them as obtuse-angled triangles or just obtuse triangles for short. The square of the addition of two triangle sides is less than that of the square of the triangle’s longest side because an obtuse angle triangle has an acute angle. Since the sides of the triangle **ABC** have the measurements **48** degrees, **112** degrees, and** 20** degrees, respectively, and a is the longest side.

**Obtuse Angles in Polygons**

A polygon with obtuse angles can be classified into **two** distinct subtypes: those that have** just** obtuse angles and those that have a **minimum** of **one** acute angle. The former category is more typical, whereas the latter category is notoriously challenging to put together.

In a polygon with obtuse angles, the sum of the individual angles will almost never add up to less than **180** degrees. This is due to the fact that the combined degrees of a triangle’s angles equal **180 **°, and a polygon with an obtuse angle must contain a minimum of one triangle.

A polygon with obtuse interior angles will have interior angles, all of which are under **180°**. This is due to the fact that an obtuse angle has a degree measurement that is more than **90°** but lower than **180°**.

Every one of the obtuse angle polygons’ outside angles is larger than **180°**. This is due to the fact that the total of all the angles in the polygon is always **360°**, and a polygon with an obtuse angle always contains at least one angle that faces the outside of the shape.

### Example 1

Show the obtuse angle in a triangle.

### Solution

Here in the figure below, a triangle **ABC** is shown.

This scalene triangle has all unequal sides, so the angles are also unequal. It can be clearly seen that at point **A,** the angle is **48°.** Likewise, at point **C,** the angle is **20°.** Both these angles are less than **90°,** so these are acute angles. In contrast to this, at point **B,** the angle is **112°** which is greater than **90°** but less than **180°.** Therefore, this is the **o****btuse angle**.

### Example 2

Show that a regular polygon has all obtuse interior angles.

### Solution

In this example, a regular hexagon is shown, which has six equal angles, each of **120**°. The exterior angle of each side of the hexagon is **60°, and the interior angle is 120°. All the angles are 120° which is obtuse because it is greater than 90° and less than 180°, so a regular hexagon has all obtuse angles present inside it.**

*All the figures above are created on GeoGebra.*

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