Angles - Explanation and Study Guide

What is an Angle?

An angle is a geometric figure formed by two rays or lines that extend from a common point, called the vertex.

Types of Angles

There are several types of angles based on their measure and relationship with each other:

• Acute Angle: An angle that measures between 0 and 90 degrees.
• Right Angle: An angle that measures exactly 90 degrees.
• Obtuse Angle: An angle that measures between 90 and 180 degrees.
• Straight Angle: An angle that measures exactly 180 degrees.
• Reflex Angle: An angle that measures between 180 and 360 degrees.

Angle Notation

Angles are typically denoted using the symbol ∠ followed by three points, with the middle point representing the vertex. For example, ∠ABC represents an angle with vertex at point B.

Angle Measurement

Angles are measured in degrees, with a full rotation around a point measuring 360 degrees. Degrees can be further divided into minutes (') and seconds (").

Angle Relationships

When two lines intersect, they form pairs of angles with specific relationships:

• Vertical Angles: Two non-adjacent angles formed by intersecting lines. They are congruent (have the same measure).
• Adjacent Angles: Two angles that share a common vertex and side, but do not overlap. Their measures add up to 180 degrees (form a linear pair) in a straight line.
• Complementary Angles: Two angles whose measures add up to 90 degrees.
• Supplementary Angles: Two angles whose measures add up to 180 degrees.

Angle Properties and Formulas

There are several important properties and formulas related to angles, including:

• Sum of Interior Angles of a Polygon: The sum of the interior angles of a polygon with n sides is given by (n-2) * 180 degrees.
• Exterior Angle of a Polygon: The exterior angle of a polygon is equal to the sum of the two remote interior angles.
• Angles in a Triangle: The sum of the interior angles of a triangle is always 180 degrees.

Angle Measurement Tools

There are various tools used to measure and draw angles, including protractors, angle rulers, and geometric software applications.

Practice Problems

Now that you have learned about angles, it's time to practice! Try solving the following problems:

1. Find the complement of an angle measuring 25 degrees.
2. Determine the type of angle formed by the hands of a clock at 3:00 PM.
3. Calculate the measure of each interior angle of a regular hexagon.

Good luck with your angle practice!