Describing Angles

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Concept

Angle

When two rays share an endpoint, an angle is formed. Below, the rays BA\overrightarrow{BA} and BC\overrightarrow{BC} create angle ABC.ABC.

When referring to a specific angle, the notation \angle is often used to name the angle. Additionally, mm\angle 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 00 ^\circ shows no rotation, while an angle of 360360 ^\circ shows a complete rotation.

Sometimes, the space outside an angle — rather than inside — is of interest. In figure above, α\alpha measures the interior angle formed by the rays, whereas β\beta 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 0.0. 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 145,145, according to the inner orientation. Therefore, the angle measures 145.145^\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\angle A and C\angle C are congruent, which can be written as AC.\angle A \cong \angle C.

Concept

Types of Angles

Angles can be classified by their measures. For angles between 00^\circ and 180,180^\circ, they can be divided into four categories: acute, right, obtuse, and straight. acute angle:measures less than 90.right angle:measures exactly 90.obtuse angle:measures greater than 90.straight angle:measures exactly 180.\begin{aligned} \textbf{acute angle} &: \text{measures less than } 90 ^\circ. \\ \textbf{right angle} &: \text{measures exactly } 90 ^\circ. \\ \textbf{obtuse angle} &: \text{measures greater than } 90 ^\circ. \\ \textbf{straight angle} &: \text{measures exactly } 180 ^\circ. \\ \end{aligned}
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 9090 ^\circ are called complementary angles.

In the figure above, CBD\angle CBD is complementary to DBA\angle DBA because

mCBD+mDBA=90. m\angle CBD + m\angle DBA = 90 ^\circ.
Concept

Supplementary Angles

Two or more angles whose measures add to equal 180180^\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\angle \alpha and β\angle \beta are supplementary because

mα+mβ=180. m\angle \alpha + m\angle \beta = 180 ^\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. \text{vertical angles are congruent.} In the figure below, this means that 13\angle 1 \cong \angle 3 and 24.\angle 2 \cong \angle 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\angle 1 and 3,\angle 3, in the figure below, have the same measure.

Notice that 1\angle 1 and 2\angle2 are supplementary because they form a straight angle. Thus, since m1+m2=180,m\angle 1+m\angle 2=180^\circ, m2m\angle 2 can be expressed as follows. m2=180m1. m\angle 2=180^\circ-m\angle 1.

In the same way, since 2\angle 2 and 3\angle 3 are supplementary, m2m\angle 2 can be expressed in another way. m2=180m3.m\angle2=180^\circ-m\angle3. By transitivity, the equations for m2m\angle 2 can be set equal to each other. 180m1=180m3. 180^\circ-m\angle1=180^\circ-m\angle3. By simplifying, the following equality yields. m1=m3. m\angle1=m\angle3. Thus, 13.\angle 1 \cong \angle 3. Therefore, vertical angles are congruent.
This reasoning can be summarized in a flowchart proof.

Exercise

In the figure, 3\angle 3 and 4949^\circ are complementary. Determine the measures of angles 1,1, 2,2, and 3.3.

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

We'll find the measure of each angle in numerical order. Since 1\angle 1 and the marked 6060^\circ 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 m1=60.m\angle 1=60 ^\circ. 1\angle 1 and 2\angle 2 are supplementary since, together, they form a straight angle. The measures of supplementary angles always add to equal 180,180 ^\circ, so m1+m260+m2=180m2=120. m\angle 1 + m\angle 2 \quad \Leftrightarrow \quad 60 ^\circ + m\angle 2 = 180 ^\circ \quad \Leftrightarrow \quad m\angle 2 = 120 ^\circ. We know that 3\angle 3 and 4949^\circ are complementary.

That means the sum of their measures is 90.90^\circ. m3+49=90m3=41. m\angle 3 + 49^\circ = 90^\circ \quad \Leftrightarrow \quad m\angle 3 = 41^\circ. Thus, 3\angle 3 is 41.41^\circ. In conclusion, m1=60,m2=120,andm3=41. m\angle 1=60^\circ, \quad m\angle 2=120^\circ, \quad \text{and} \quad m\angle 3 = 41^\circ.

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Exercises

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