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Geometry Basics

Describing Angles

Concept

Angle

An angle is the set of points in the plane formed by two different rays that share the same starting point. This common point is called the vertex of the angle and the rays are the sides of the angle.

Two Rays with the same starting point

There are different ways to denote an angle and all involve the symbol in front of the name. For the second notation, the vertex always must be in the middle.

Using the Vertex Using the Vertex and One Point on each Ray Using a Number
or

The measure of an angle, denoted by is the number of degrees between the rays, found by applying the Protractor Postulate. When two angles have the same measure, they are said to be congruent.

Extra

Interior and Exterior of an Angle

An angle divides the plane into two parts.

  • The region between the sides, or interior of the angle.
  • The region outside the sides, or exterior of the angle.
Interior and Exterior of an angle
Notice that the interior of an angle is the region for which the angle measure is less than
Method

Measuring an Angle with a Protractor

A protractor is a tool that can be used to measure angles in degrees.

The measure of the angle below will be found using a protractor.

To determine the measure of an angle, one of its rays — it doesn't matter which — is placed along the bottom of the protractor so that it is aligned with The intersection of the rays is placed in the middle, as shown.

The ray intersects the zero on the right side of the protractor, which means the inner orientation, which increases in the counterclockwise direction, is used to determine the angle.

The other ray is located at according to the inner orientation. Therefore, the angle measures

Construction

Copying an Angle

unfold_more

An angle can be constructed as a copy of a given angle using a compass. To begin, place the sharp end of the compass at the vertex of the angle. Then, draw an arc across the rays any distance from the vertex.

Next, sketch an approximation of one of the rays — here that is the horizontal ray. Without changing the settings of the compass, position the sharp end at the endpoint of the copied ray, and draw an arc.

Next, adjust the compass to measure the distance between the rays at their points of intersection with the arc on the original angle. Keeping this measurement, align the sharp point with the arc's point of intersection with the horizontal ray on the copy. Mark this distance on the other end of the arc.

On the copy, the second ray can be drawn from the endpoint on the first ray. Draw a segment from this point through the marked position on the arc.

info
Concept

Angle Bisector

An angle bisector is a ray that divides an angle into two angles that have the same measure.

Construction

Bisecting an Angle

unfold_more

An angle can be bisected using a compass.

Start by placing the sharp end of the compass at the vertex of the angle. Draw an arc across the rays.

Next, keeping the compass set, place its sharp end at the point where one ray intersects the arc. Draw a new arc.

Do the same with the other ray, making sure this arc intersects the previous arc.

Use a ruler or a straightedge to draw a ray from the vertex through the intersection between the two arcs.

This ray is the angle bisector, and divides the angle into two angles of equal measure.

info
Concept

Congruent Angles

Angles that have the same measure are said to be congruent. In a figure, congruent angles are usually indicated by the same number of arcs. In the figure below, and are congruent, which can be written as

Concept

Types of Angles

Angles can be classified by their measures. For angles between and they can be divided into four categories: acute, right, obtuse, and straight.
Concept

Pairs of Angles

Pairs of angles can be classified in different ways depending on how their measures relate.

Concept

Complementary Angles

Two angles whose measures add to are called complementary angles.

In the figure above, is complementary to because

Concept

Supplementary Angles

Two or more angles whose measures add to equal are called supplementary angles. Two adjacent supplementary angles are called a linear pair, and together they form a straight angle.

In the figure above, and are supplementary because

Concept

Vertical Angles

When two lines or line segments intersect, vertical angles are formed on opposite sides of the point of intersection. In the figure below, vertical angles are marked with the same number of hatch marks.

Theory

Vertical Angles Theorem

The Vertical Angles Theorem states that In the figure below, this means that and

This can be proven using supplementary angles.

Proof

Vertical Angles Theorem
Proof

Vertical Angles Theorem

To prove that vertical angles are congruent, it will be shown that and in the figure below, have the same measure.

Notice that and are supplementary because they form a straight angle. Thus, since can be expressed as follows.

In the same way, since and are supplementary, can be expressed in another way. By transitivity, the equations for can be set equal to each other. By simplifying, the following equality yields. Thus, Therefore, vertical angles are congruent.
This reasoning can be summarized in a flowchart proof.

fullscreen
Exercise

In the figure, and are complementary. Determine the measures of angles and

Four straight lines with given and unknown angles marked at the points of intersection
Show Solution
Solution

We'll find the measure of each angle in numerical order. Since and the marked angle are on opposite sides of a point of intersection, they form vertical angles.

Two straight lines with given and unknown angles marked at the point of intersection

Vertical angles are always congruent, so and are supplementary since, together, they form a straight angle. The measures of supplementary angles always add to equal so We know that and are complementary.

That means the sum of their measures is Thus, is In conclusion,

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