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The measure of an angle is the distance between two lines which are intersecting at a point. There are names given to some angles of a particular value. For example, the right angle for angles of 90 degrees and the straight angle for values of angle Although the study of angles seems simple and trivial, it opens the path to the different aspects of architecture. Even carpenters use angles to help create their work. Even when determining the trajectory of airplanes or launching missiles, the accuracy of the angle is very important....Read MoreRead Less

If a circle is divided into 360 equal parts, the angle that turns \(\frac{1}{360}\) th of a circle is \(1^{\circ}\). A full turn around the circle is 360 degrees. In the image shown below, each of the smallest divisions is a degree.

The distance between the intersecting lines of the two rays provided below is \(\frac{60}{360}\). What is the angle between the two rays?

From the theory elaborated previously, we know that the angle that turns \(\frac{1}{360}\) th of a circle is \(1^{\circ}\). This means that the angle that turns \(\frac{60}{360}\) th of a circle should be 60°.

**Example 1:**

**Examine the two representations given below and explain if both the representations show the same angles.**

**Solution :**

The first picture is a representation of 360 degrees and the second picture is a representation of 0 degrees. The value of the angle is different in both these representations.

**Example 2:**

Find the measure of the angle given below.

**Solution :**

We know that \(\frac{1}{360}\) th of a circle is one degree.

This means that \(\frac{57}{360}\) th is 57 degrees.

Therefore, the measure of the angle given here is 57 degrees.

**Example 3:**

Find the measure of the angle given below.

We know that \(\frac{1}{360}\) th of a circle is one degree.

\(\frac{1}{10}\) th of a circle can be calculated by multiplying \(\frac{1}{10}\) with 360.

\(=\frac{1}{10} \times 360=36^{\circ}\)

Therefore, \(\frac{1}{10}\) th of a circle is 36 degrees.

**Example 4:**

Categorize the given set of angles as acute, straight, right or obtuse.

A. 25°

B. 150°

C. 90°

D. 180°

**Solution:**

A. 25°: 25 degrees can be categorized as an acute angle as it is less than 90 degrees.

B. 150°: 150 degrees can be classified as an obtuse angle as it is greater than 90 degrees.

C. 90°: 90 degrees is also called the right angle.

D. 180°: is also called a straight angle as the resultant angle forms a straight line.

**Example 5:**

**Tammy and Lola ordered a chocolate cake. Lola took \(\frac{3}{7}\) th of the cake and \(\frac{4}{9}\) th ****was taken by Tammy. Who took the larger slice of cake?**

**Solution:** ** **

We can find out who took the larger slice of cake by calculating the angle at the center of the cake.

** **

To find the angle we can find the \(\frac{3}{7}\) th and \(\frac{4}{9}\) th of 360 degrees.

\(\frac{3}{7}\) th of 360 degrees = \(\frac{3}{7}\times 360\) = 154.29°

\(\frac{4}{9}\) th of 360 degrees = \(\frac{4}{9}\times 360\) = 160°

The area covered by the 160 degree angle is greater than 154.29°.

This shows that Tammy took more cake than Lola.

**Example 6:**

**A tire has five spokes fashioned on the rim which divides the tire into five equal parts. Find the angle measure of each part.**

**Solution:**

A complete circle is 360 degrees, to find the angle of one part out of 5, 360 degrees needs to be divided by 5.

The measure of the angle between each section divided by the spokes is \(=\frac{360}{5}=72^{\circ}\)

**Example 7:**

**A pizza was cut into 12 equal pieces. Out of these three pieces were eaten. What is the angle measure of the three pieces which were eaten?**

**Solution:**

To find the angle measure of each slice, we need to divide 360 by 12.

The measure of each slice \(=\frac{360}{12}=30^{\circ}\)

To find the measure of three out of the twelve slices, all that needs to be done is to multiply the angle measure of one slice by three.

\(=30\times 3 = 90^{\circ}\)

Therefore, the angle measure of three slices is 90°.

Frequently Asked Questions

An angle is a measure of rotation from a given point. Although the change between two angles looks small, angle measures are used when launching missiles or spacecraft. A small change could determine the change of trajectory by miles. Engineers use angles to plan the different structures they create. There are many other examples of the use of angles and you could try to find some of these use cases.

The angle between the two rays of a right angle is 90 degrees, whereas the angle between the two rays for a straight angle is 180 degrees. Resultantly, a straight angle will be a straight line and a right angle will be ‘L-shaped’

No, even temperature is measured in degrees.

Yes, values of angles can be greater than 360 degrees, although they will be rotating about the same point.

The most basic device that is used to measure angles is a protractor. A compass, hand squares and set squares can also be used to measure angles. More sophisticated tools include navigation plotters, sextants, theodolites, miter saws, goniometers and inclinometers.