Describing Angles

Concept

Angle

When two rays share an endpoint, an angle is formed. Below, the rays $B A$ and $B C$ create angle $A B C .$

When referring to a specific angle, the notation $∠$ is often used to name the angle. Additionally, $m ∠$ is used to note the size or measure of an angle, which can be interpreted as the amount of turn in the rotation between the rays. It is often measured in degrees, but can also be measured in radians. An angle of $0^{\circ}$ shows no rotation, while an angle of $36 0^{\circ}$ shows a complete rotation.

Sometimes, the space outside an angle — rather than inside — is of interest. In figure above,

α

measures the interior angle formed by the rays, whereas

β

measures the exterior angle.

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 $0 .$ 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

145,

according to the inner orientation. Therefore, the angle measures

14 5^{\circ} .

Construction

Copying an Angle

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.

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

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.

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, $∠ A$ and $∠ C$ are congruent, which can be written as $∠ A ≅ ∠ C .$

Concept

Types of Angles

Angles can be classified by their measures. For angles between

0^{\circ}

and

18 0^{\circ},

they can be divided into four categories: acute, right, obtuse, and straight.

acute angle right angle obtuse angle straight angle : measures less than 9 0^{\circ} . : measures exactly 9 0^{\circ} . : measures greater than 9 0^{\circ} . : measures exactly 18 0^{\circ} .

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 $9 0^{\circ}$ are called complementary angles.

In the figure above, $∠ C B D$ is complementary to $∠ D B A$ because

m ∠ C B D + m ∠ D B A = 9 0^{\circ} .

Concept

Supplementary Angles

Two or more angles whose measures add to equal

18 0^{\circ}

are called supplementary angles. Two adjacent supplementary angles are called a linear pair, and together they form a straight angle.

In the figure above, $∠ α$ and $∠ β$ are supplementary because

m ∠ α + m ∠ β = 18 0^{\circ} .

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 $vertical angles are congruent .$ In the figure below, this means that $∠ 1 ≅ ∠ 3$ and $∠ 2 ≅ ∠ 4 .$

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 $∠ 1$ and $∠ 3,$ in the figure below, have the same measure.

Notice that $∠ 1$ and $∠ 2$ are supplementary because they form a straight angle. Thus, since $m ∠ 1 + m ∠ 2 = 18 0^{\circ},$ $m ∠ 2$ can be expressed as follows. $m ∠ 2 = 18 0^{\circ} - m ∠ 1 .$

In the same way, since $∠ 2$ and $∠ 3$ are supplementary, $m ∠ 2$ can be expressed in another way. $m ∠ 2 = 18 0^{\circ} - m ∠ 3 .$ By transitivity, the equations for $m ∠ 2$ can be set equal to each other. $18 0^{\circ} - m ∠ 1 = 18 0^{\circ} - m ∠ 3 .$ By simplifying, the following equality yields. $m ∠ 1 = m ∠ 3 .$ Thus, $∠ 1 ≅ ∠ 3 .$ Therefore, vertical angles are congruent.
This reasoning can be summarized in a flowchart proof.

Determine the measures of the angles

fullscreen

Exercise

In the figure, $∠ 3$ and $4 9^{\circ}$ are complementary. Determine the measures of angles $1,$ $2,$ and $3 .$

Show Solution

Solution

We'll find the measure of each angle in numerical order. Since $∠ 1$ and the marked $6 0^{\circ}$ angle are on opposite sides of a point of intersection, they form vertical angles.

Vertical angles are always congruent, so $m ∠ 1 = 6 0^{\circ} .$ $∠ 1$ and $∠ 2$ are supplementary since, together, they form a straight angle. The measures of supplementary angles always add to equal $18 0^{\circ},$ so $m ∠ 1 + m ∠ 2 \Leftrightarrow 6 0^{\circ} + m ∠ 2 = 18 0^{\circ} \Leftrightarrow m ∠ 2 = 12 0^{\circ} .$ We know that $∠ 3$ and $4 9^{\circ}$ are complementary.

That means the sum of their measures is $9 0^{\circ} .$ $m ∠ 3 + 4 9^{\circ} = 9 0^{\circ} \Leftrightarrow m ∠ 3 = 4 1^{\circ} .$ Thus, $∠ 3$ is $4 1^{\circ} .$ In conclusion, $m ∠ 1 = 6 0^{\circ}, m ∠ 2 = 12 0^{\circ}, and m ∠ 3 = 4 1^{\circ} .$

Construction

Copying an Angle

Construction

Bisecting an Angle

Proof

Example

Determine the measures of the angles