Obtuse Angles

In geometry, angles are the basic concepts used to understand how lines, shapes, and spaces relate to each other. Among the different kinds of angles, the obtuse angle is one of the most important classifications. This kind of angle is essential in theoretical geometry as well as in practical applications, ranging from the construction of buildings to the arrangement of daily objects. So, let’s explore the properties, formulas, and measurement techniques for obtuse angles. 

1.0What is an Obtuse Angle?

The word angles, in geometry, comes from the Latin word “angulus”, which simply means corner. There are different types of angles in two-dimensional geometry, and obtuse angles are the most important type among these angles. The degrees of an obtuse angle are always greater than a right angle or 90° and less than a straight angle or 180°. In simple words, any angle between 90° and 180° is classified as an obtuse angle. However, note that angles greater than 180° are termed the reflex angle and are different from obtuse angles. 

2.0Properties of Obtuse Angles

Obtuse angles are different from other types of angles due to certain properties these angles possess, which include: 

• Measurement: As stated, an obtuse angle is any angle between 90° and 180°. Thus, the measure of an obtuse angle is always between 91° and 179°.
• Sum of Angles: Whenever an obtuse angle is combined with an acute angle (which is an angle smaller than 90°), their sum would always be equal to 180°. Like, an obtuse angle measuring 120° and an acute angle measuring 60° combine to make a total of 180° (120° + 60° = 180°).
• Not Complementary: Acute angles (which add up to 90°) are not complementary, but obtuse angles are not complementary because their measure is already greater than 90°. Complementary angles need to add up to 90°. 

3.0Measuring an Obtuse Angle 

An obtuse angle, or any angle, is generally measured with the help of a protractor, a D-shaped mathematical tool used to measure angles in geometric figures. Follow the below-mentioned steps for measuring an obtuse angle with ease and accuracy: 

• Position the centre of the protractor over the vertex of the angle.
• Position one of the rays over the zero-degree line of the protractor.
• Read the number at which the second ray crosses the scale of the protractor. If it is between 90° and 180°, you have an obtuse angle.

4.0Acute and Obtuse Angles 

Acute and obtuse angles are the two aspects of two-dimensional geometry and topics related to lines and rays. However, both of these angles possess certain important differences, which include: 

Acute Angle	Obtuse Angle
Acute angles are the ones that measure less than 90°.	Degrees of an obtuse angle generally measure greater than 90° and less than 180°
Visually, these angles appear to be sharp and narrow at their ends.	Obtuse angles appear wide and spread out at the meeting point of two rays.
Acute angles can be complementary (the sum of a pair of angles equal to 90°) with another acute angle.	These angles can never form a complementary angle pair with another obtuse angle.
The range of acute angles always lies between 0° to 90°.	The range of obtuse angles starts from 90° and ends at 180°.
Examples of acute angles can be 45°, 60°, and even 89°.	Examples of obtuse angles are 120°, 91°, and 179°.

5.0Triangle with an Obtuse Angle

A triangle with an obtuse angle is a triangle with one angle equal to greater than 90° while the other two angles equal to less than 90°. The sum of these angles always equals 180°, also known as the angle sum property of a triangle. For example, if in a given triangle ABC, one angle is equal to ∠A ＞90° and ∠B and ∠C ＜90°, such that ∠A + ∠B + ∠C = 180°. 

 It's worth mentioning that any triangle that has one obtuse angle will always be an obtuse triangle, no matter what the actual measurements of the other two angles are, as long as they are acute.

6.0Real-Life Application of Obtuse Angles 

Obtuse angles are not merely theoretical entities; they occur in numerous real-life scenarios. These are some familiar examples:

• Clock Hands: At 10:10 or 2:50, the angle between the hour hand and the minute hand of a clock is an obtuse angle. The angle is greater than 90° but less than 180°.
• Scissors: The angle between the handles of open scissors is an obtuse angle.
• Ceiling Fan Blades: When the blades of a ceiling fan are opened up, they tend to create an obtuse angle.
• Road Intersections: In city planning, certain road intersections can create obtuse angles where roads intersect at a broader angle than a right angle.

7.0Solved Examples: Obtuse Angle 

Problem 1: In an obtuse triangle, one angle measures 100°, and the other angle measures 40°. What is the third angle?

Solution: Given that 100° and 40° are the angles of an obtuse triangle. For calculating the third angle, say x, use the angle sum property of a triangle. 

100° + 40° + x = 180°

140° + x = 180°

x = 180° – 140° = 40° 

Problem 2: In a triangle, two angles measure 50° and 60°. What is the measure of the third angle? Also, mention the type of this triangle. 

Solution: Let the third angle be x. By the angle sum property of a triangle, we know

50° + 60° + x = 180°

110° + x = 180°

x = 180° – 110° = 70°. 

It is an acute angle triangle.

Problem 3: You are given three angles: 110°, 50°, and 60°. Verify if these angles can form a triangle.

Solution: To verify if the given angles form a triangle, use the angle sum property of a triangle. According to the question, 

110° + 50° + 60° = 220°

220°＞180°

Hence, the given angles don’t form a triangle.