Obtuse Angle

Obtuse Angle is a type of angle that is more than a right angle (90°) but less than a straight angle (180°). Unlike acute angles, which are smaller than 90° and right angles, which measure exactly 90°, an obtuse angle exhibits an opening wider than a right angle.

Consider a corner, then imagine one side moving away from the other, making the angle wider. It's not wide enough to form a straight line, but wider than a right angle. In this article, we will be discussing all the things related to Obtuse Angle including example formula as well it's comparison with acute and right angle as well.

What is an Obtuse Angle?

An obtuse angle is a type of angle in geometry that measures more than 90° but less than 180°. Visually, it appears wider than a right angle (90°) but narrower than a straight angle (180°). In simpler terms, an obtuse angle is "more open" than a right angle.

Obtuse Angle Definition

The angle which measure greater than 90° and less than 180° is called Obtuse Angle

Obtuse Angle Formula

An obtuse angle symbolized by spans beyond a right angle (measuring 90°) but falls short of forming a straight line (measuring 180°). It occupies a range greater than 90° yet less than 180° 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° distinct from acute angles and straight angles.

90° < θ < 180°

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 Example

An obtuse angle, typically found in triangles and those triangles are called obtuse-angled triangle. For instance, if one angle within a triangle measures 120° it indicates an angle that surpasses the 90° threshold yet doesn't extend to the full 180°. This scenario illustrates an obtuse angle within the triangle, representing an angle greater than 90° but less than 180°.

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 Triangle

A Triangle whose one of the angle is greater than 90° but less than 180° is called Obtuse Angled Triangle. In a triangle we can have only one obtuse angle.

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° but less than 180°. 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°.

Right, Acute, Obtuse Angle

The angle less than 90° is called acute angle. The angle equals to 90° is called right angle. The angle greater than 90° and less than 180° is called obtuse angle.

Characteristics	Definition	Diagram
Right Angle	The angle that is equal to 90° is called right angle.
Acute Angle	The angle that is less than 90° is called acute angle.
Obtuse Angle	The angle that is greater than 90° is called acute angle.

Read More,

• Types of Angles
• Lines and Angles
• Pairs of Angles 

Solved Examples on Obtuse Angle

Example 1: In triangle PQR, if angle P measures 60° and angle Q measures 45°, determine the measure of angle R.

Solution:

For triangle PQR, the sum of its interior angles equals 180°. Using this:

Angle P + Angle Q + Angle R = 180°

Given that Angle P = 60° and Angle Q = 45°:

60° + 45° + Angle R = 180°

⇒ 105° + Angle R = 180°

⇒ Angle R = 180° - 105°

⇒ Angle R = 75°

Example 2: The exterior angle of a polygon is measured at 130°. Calculate the corresponding interior angle.

Solution:

For any polygon, the exterior angle plus its corresponding interior angle forms a straight line, which measures 180°. Therefore:

Exterior angle + Interior angle = 180°

Given the exterior angle = 130°:

130° + Interior angle = 180°

⇒ Interior angle = 180° - 130°

⇒ Interior angle = 50°

Example 3: If an angle measures 160°, find its complement.

Solution:

The complement of an angle is the difference between the angle and 90°.

Given angle = 160°:

Complement = 90° - 160°

⇒ Complement = -70°

However, as a complement can't be negative, it implies that the given angle of 160° is an obtuse angle itself and doesn't have a complement within the defined range of angles (0 to 90°).

Example 4: In a quadrilateral, one of the angles measures 125°. Is this angle obtuse?

Solution:

Yes, an angle measuring 125° is greater than 90°, so it is classified as an obtuse angle.

Example 5: An angle measures 150°. 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° an angle measuring 150° fits into the range and can be classified as obtuse.

Example 6: In a pentagon one angle measures 110° and another measures 95°. Are these angles obtuse?

Solution:

Among the two angles given the 110-degree angle is an obtuse angle as it is larger than 90° whereas the 95-degree angle is acute as it's less than 90°.

Example 7: Within a hexagon, an angle measures 135°. Does this angle fit the description of an obtuse angle?

Solution:

Yes, an angle that measures 135° surpasses 90° yet remains less than 180°, aligning with the characteristics of an obtuse angle.

Example 8: If an angle measures 170°, would it qualify as an obtuse angle?

Solution:

Absolutely, an angle that measures 170° falls within the range of obtuse angles since it exceeds 90° but remains below 180°. 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°, 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.