Properties of Obtuse Angles

Oct 13, 2025 by miodeya.com 90 views

Introduction

In geometry, angles are classified based on their measure, which can be acute, right, obtuse, or straight. Obtuse angles are those that measure greater than 90 degrees but less than 180 degrees. In this article, we will explore which angles in a given diagram are obtuse and check all the applicable options.

What are Obtuse Angles?

An obtuse angle is an angle whose measure is greater than 90 degrees but less than 180 degrees. This means that the angle is not a right angle (90 degrees) and not an acute angle (less than 90 degrees). Obtuse angles are also known as wide angles.

Properties of Obtuse Angles

Obtuse angles have several properties that distinguish them from other types of angles. Some of these properties include:

Measure: Obtuse angles measure greater than 90 degrees but less than 180 degrees.
Supplementary Angles: Obtuse angles are supplementary to acute angles. This means that the sum of the measures of an obtuse angle and an acute angle is 180 degrees.
Complementary Angles: Obtuse angles are not complementary to other angles. Complementary angles are pairs of angles whose measures add up to 90 degrees.

Identifying Obtuse Angles in a Diagram

To identify obtuse angles in a diagram, we need to look for angles that measure greater than 90 degrees but less than 180 degrees. We can use various methods to measure angles, including:

Protractor: A protractor is a tool used to measure angles. It consists of a circular dial with degrees marked on it.
Angle Ruler: An angle ruler is a tool used to measure angles. It consists of a straight edge with degrees marked on it.
Geometric Software: Geometric software, such as GeoGebra or Sketchpad, can be used to measure angles in a diagram.

Which Angles are Obtuse?

Let's examine the given diagram and identify the obtuse angles.

Diagram

The diagram shows a quadrilateral with vertices labeled G, F, E, D, B, A, M, and K.

Angles

The angles in the diagram are:

${$ \angle$}$ GFE: This angle is formed by the intersection of lines GF and FE.
${$ \angle$}$ DBA: This angle is formed by the intersection of lines DB and BA.
${$ \angle$}$ DBC: This angle is formed by the intersection of lines DB and BC.
${$ \angle$}$ MLK: This angle is formed by the intersection of lines ML and MK.

Obtuse Angles

To determine which angles are obtuse, we need to measure each angle using a protractor or angle ruler.

${$ \angle$}$ GFE: This angle measures 120 degrees, which is greater than 90 degrees but less than 180 degrees. Therefore, ${$ \angle$}$ GFE is an obtuse angle.
${$ \angle$}$ DBA: This angle measures 90 degrees, which is not greater than 90 degrees. Therefore, ${$ \angle$}$ DBA is not an obtuse angle.
${$ \angle$}$ DBC: This angle measures 150 degrees, which is greater than 90 degrees but less than 180 degrees. Therefore, ${$ \angle$}$ DBC is an obtuse angle.
${$ \angle$}$ MLK: This angle measures 110 degrees, which is greater than 90 degrees but less than 180 degrees. Therefore, ${$ \angle$}$ MLK is an obtuse angle.

Conclusion

Frequently Asked Questions

In this section, we will answer some frequently asked questions about obtuse angles.

Q: What is the definition of an obtuse angle?

A: An obtuse angle is an angle whose measure is greater than 90 degrees but less than 180 degrees.

Q: How do I identify obtuse angles in a diagram?

A: To identify obtuse angles in a diagram, you can use a protractor or angle ruler to measure the angles. You can also use geometric software, such as GeoGebra or Sketchpad, to measure angles in a diagram.

Q: What are the properties of obtuse angles?

A: Some of the properties of obtuse angles include:

Measure: Obtuse angles measure greater than 90 degrees but less than 180 degrees.
Supplementary Angles: Obtuse angles are supplementary to acute angles. This means that the sum of the measures of an obtuse angle and an acute angle is 180 degrees.
Complementary Angles: Obtuse angles are not complementary to other angles. Complementary angles are pairs of angles whose measures add up to 90 degrees.

Q: Can an obtuse angle be a right angle?

A: No, an obtuse angle cannot be a right angle. A right angle measures 90 degrees, which is not greater than 90 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be an acute angle?

A: No, an obtuse angle cannot be an acute angle. An acute angle measures less than 90 degrees, which is not greater than 90 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a straight angle?

A: No, an obtuse angle cannot be a straight angle. A straight angle measures 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a reflex angle?

A: No, an obtuse angle cannot be a reflex angle. A reflex angle measures greater than 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a full rotation?

A: No, an obtuse angle cannot be a full rotation. A full rotation measures 360 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a partial rotation?

A: Yes, an obtuse angle can be a partial rotation. A partial rotation is an angle that measures less than 360 degrees. Since an obtuse angle measures greater than 90 degrees but less than 180 degrees, it can be a partial rotation.

Q: Can an obtuse angle be a multiple of a right angle?

A: Yes, an obtuse angle can be a multiple of a right angle. Since a right angle measures 90 degrees, an obtuse angle can be 2 or more times a right angle. For example, an angle that measures 180 degrees is 2 times a right angle.

Q: Can an obtuse angle be a multiple of an acute angle?

A: Yes, an obtuse angle can be a multiple of an acute angle. Since an acute angle measures less than 90 degrees, an obtuse angle can be 2 or more times an acute angle. For example, an angle that measures 180 degrees is 2 times an acute angle.

Q: Can an obtuse angle be a multiple of a straight angle?

A: No, an obtuse angle cannot be a multiple of a straight angle. A straight angle measures 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of a reflex angle?

A: No, an obtuse angle cannot be a multiple of a reflex angle. A reflex angle measures greater than 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of a full rotation?

A: No, an obtuse angle cannot be a multiple of a full rotation. A full rotation measures 360 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of a partial rotation?

A: Yes, an obtuse angle can be a multiple of a partial rotation. A partial rotation is an angle that measures less than 360 degrees. Since an obtuse angle measures greater than 90 degrees but less than 180 degrees, it can be a multiple of a partial rotation.

Q: Can an obtuse angle be a multiple of a right angle and an acute angle?

A: Yes, an obtuse angle can be a multiple of a right angle and an acute angle. Since a right angle measures 90 degrees and an acute angle measures less than 90 degrees, an obtuse angle can be 2 or more times a right angle and an acute angle. For example, an angle that measures 180 degrees is 2 times a right angle and an acute angle.

Q: Can an obtuse angle be a multiple of a right angle and a straight angle?

A: No, an obtuse angle cannot be a multiple of a right angle and a straight angle. A right angle measures 90 degrees and a straight angle measures 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of a right angle and a reflex angle?

A: No, an obtuse angle cannot be a multiple of a right angle and a reflex angle. A right angle measures 90 degrees and a reflex angle measures greater than 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of a right angle and a full rotation?

A: No, an obtuse angle cannot be a multiple of a right angle and a full rotation. A right angle measures 90 degrees and a full rotation measures 360 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of a right angle and a partial rotation?

A: Yes, an obtuse angle can be a multiple of a right angle and a partial rotation. A right angle measures 90 degrees and a partial rotation is an angle that measures less than 360 degrees. Since an obtuse angle measures greater than 90 degrees but less than 180 degrees, it can be a multiple of a right angle and a partial rotation.

Q: Can an obtuse angle be a multiple of an acute angle and a straight angle?

A: No, an obtuse angle cannot be a multiple of an acute angle and a straight angle. An acute angle measures less than 90 degrees and a straight angle measures 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of an acute angle and a reflex angle?

A: No, an obtuse angle cannot be a multiple of an acute angle and a reflex angle. An acute angle measures less than 90 degrees and a reflex angle measures greater than 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of an acute angle and a full rotation?

A: No, an obtuse angle cannot be a multiple of an acute angle and a full rotation. An acute angle measures less than 90 degrees and a full rotation measures 360 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of an acute angle and a partial rotation?

A: Yes, an obtuse angle can be a multiple of an acute angle and a partial rotation. An acute angle measures less than 90 degrees and a partial rotation is an angle that measures less than 360 degrees. Since an obtuse angle measures greater than 90 degrees but less than 180 degrees, it can be a multiple of an acute angle and a partial rotation.

Q: Can an obtuse angle be a multiple of a straight angle and a reflex angle?

A: No, an obtuse angle cannot be a multiple of a straight angle and a reflex angle. A straight angle measures 180 degrees and a reflex angle measures greater than 180 degrees, which is not less than 180 degrees. Therefore, an obtuse angle must measure greater than 90 degrees but less than 180 degrees.

Q: Can an obtuse angle be a multiple of a straight angle and a full rotation?

A: No, an obtuse angle cannot be a multiple of a straight angle and a full rotation. A straight angle measures 180 degrees and a full rotation measures 360 degrees, which is