What You Need to Know About Angles in Geometry

Angles are an essential component of geometry and understanding them is key to understanding the relationships between polygons, triangles, parallelograms, trapezoids, and inscribed angles. In this article, we'll go over the basics of angles and provide some practice problems to help you better understand how to use them in your math studies.

What are Angles?

An angle is defined as a figure formed by two lines (or rays) that meet at a certain point, called the vertex. The lines that form an angle are called sides of the angle. An angle can be measured in degrees or radians. The figure below illustrates an angle with its two sides and vertex.

The angle in the figure above is an acute angle, which is an angle that measures less than 90 degrees. In addition to acute angles, there are also right angles, which measure exactly 90 degrees, obtuse angles, which measure more than 90 degrees but less than 180 degrees, and straight angles, which measure exactly 180 degrees.

Types of Angles

Angles can be classified in several ways. One type of angle is called a complementary angle, which is an angle that adds up to 90 degrees. For example, if one angle measures 40 degrees, then its complementary angle measures 50 degrees. Another type of angle is called a supplementary angle, which is an angle that adds up to 180 degrees. For example, if one angle measures 90 degrees, then its supplementary angle measures 90 degrees.

Angles can also be classified according to their relative size. An obtuse angle is one that measures more than 90 degrees, while an acute angle is one that measures less than 90 degrees. An angle can also be classified as right or straight. A right angle measures exactly 90 degrees, while a straight angle measures exactly 180 degrees.

Angles in Polygons

The angles of a polygon are the angles between two adjacent sides. A triangle is a polygon with three sides, and it has three angles. A quadrilateral is a polygon with four sides, and it has four angles. A pentagon is a polygon with five sides, and it has five angles. The sum of the angles in a polygon is equal to (n-2)*180, where n is the number of sides in the polygon. For example, the sum of the angles of a triangle is 180 degrees, and the sum of the angles of a quadrilateral is 360 degrees.

Angles in Parallelograms

A parallelogram is a quadrilateral with two pairs of parallel sides. The angles of a parallelogram are all equal. This means that a parallelogram has four angles that all measure the same. For example, if one angle of a parallelogram measures 40 degrees, then all other angles of the parallelogram measure 40 degrees.

Angles in Trapezoids

A trapezoid is a quadrilateral with one pair of parallel sides. The angles of a trapezoid can be different. This means that a trapezoid can have two angles that are equal, two angles that are not equal, or four angles that are all different. For example, if one angle of a trapezoid measures 40 degrees, then the other three angles could measure 30 degrees, 50 degrees, and 90 degrees.

Inscribed Angles

An inscribed angle is an angle that is formed inside a circle. The sides of an inscribed angle are chords of the circle. The measure of an inscribed angle is equal to half the measure of the central angle that subtends (or passes through) the same arc. For example, if one inscribed angle measures 30 degrees, then the central angle that subtends the same arc measures 60 degrees.

Practice Problems

Let's try out some practice problems to test your knowledge of angles.

1. What is the sum of the angles in a pentagon?
Answer: 540 degrees.

2. What is the measure of an obtuse angle?
Answer: More than 90 degrees but less than 180 degrees.

3. What is the measure of a complementary angle?
Answer: 90 degrees.

4. What is the measure of an inscribed angle?
Answer: Half the measure of the central angle that subtends the same arc.

5. What is the measure of the angles in a parallelogram?
Answer: All equal.

6. What is the sum of the angles in a triangle?
Answer: 180 degrees.

7. What type of angle is one that measures exactly 90 degrees?
Answer: Right angle.

8. What type of angle is one that measures exactly 180 degrees?
Answer: Straight angle.

9. What type of angle is one that measures more than 90 degrees but less than 180 degrees?
Answer: Obtuse angle.

10. What type of angle is one that measures less than 90 degrees?
Answer: Acute angle.

Summary

In this article, we've gone over the basics of angles in geometry. We've discussed the types of angles, angles in polygons, angles in parallelograms, angles in trapezoids, and inscribed angles. We've also provided some practice problems to help you better understand how to use angles in your math studies. With this knowledge, you should now have a better understanding of angles and how to use them in your math studies.

FAQ

What is the difference between an acute angle and an obtuse angle?

An acute angle has a measure that is less than 90 degrees, while an obtuse angle has a measure that is greater than 90 degrees.

What is a straight angle?

A straight angle has a measure of exactly 180 degrees, and it is formed by two rays that point in opposite directions from a common endpoint.