About Measuring Angles with a Protractor
What are Angles?
When two rays are joined at their endpoints in geometry, an angle is formed. The sides or arms of the angle are the rays that meet at a point. An angle is denoted by the symbol “∠”.
Arms of the Angles
The two rays that meet at a common point to form the angle are known as the arms of the angle. Examine the diagram, which shows that the arms of the angle POQ are OP and OQ.
The Vertex of an Angle
The vertex is the point where the two rays meet at their common endpoint. Examine the diagram, where the vertex O represents the point where the two arms OP and OQ meet.
Types of Angles
Based upon the measure of the angle there are two types of angles – acute and obtuse.
Acute Angles
Acute angles are defined as those that are greater than 0 degree but less than 90 degrees.
Obtuse Angle
An obtuse angle is one that is always greater than 90 degrees but less than 180 degrees. In other words, the measure of an obtuse angle lies somewhere between 90 and 180 degrees.
How do you Measure an Angle?
Angles are measured using protractors and pattern blocks.
How to Measure Angles Using a Pattern Block?
Pattern blocks are a set of six shapes in six colors which includes orange squares, yellow hexagons, tan rhombuses, red trapezoids, blue rhombuses and green triangles.
The table below shows these standard pattern blocks with their angle measure.
Let’s take an example: Find the angle measurement using the given pattern block.
Solution:
The pattern block has a 60-degree angle at each vertex.
The angle is equal to 2 of the angles of the pattern block.
So, the measure of the angle is \( = 120^{\circ} \)
Measuring Angles by Using a Protractor
A protractor is a measuring tool. We can use a protractor to measure an angle by following these steps:
Measure of Acute angle:
Let’s try to measure the \(\angle {POQ}\) we’ve been given.
Step 1: Align the protractor with the ray OQ, as shown in the diagram below. Begin reading the inner scale at the 0 mark on the protractor’s bottom right.
Step 2: The angle is measured by the number on the protractor that coincides with the second ray. Using the protractor’s inner scale, observe the measure of the angle.
As a result, \(\angle {POQ} = 50^{\circ} \).
Measure of Obtuse angle:
Let us now measure the \(\angle {POQ}\) that has been provided.
Step 1: Align the protractor with the ray OQ, as shown in the diagram below. Begin reading the outer scale at the 0 mark on the protractor’s bottom left.
Step 2: The angle is measured by the number on the protractor that coincides with the second ray. Using the protractor’s outer scale, observe the measure of the angle.
As a result, \( \angle {POQ} = 143^{\circ} \)
How Do You Construct Angles?
An angle can be constructed using a protractor.
For example let’s construct a 60-degree angle POQ.
Step 1: Draw a ray OQ and align it with the protractor as shown.
Step 2: Using the protractor’s inner scale, mark a point P above the \( 60^{\circ} \) marking on the protractor.
Step 3: Remove the protractor and draw a ray from O to P. As a result, \(\angle POQ = 60^{\circ} \) is the required angle.
Solved Examples on Measuring Angles with a Protractor
Example 1:
Find the measure of the angle using the given pattern block.
Solution:
Each acute angle of the pattern block has a measure of \( 30^{\circ} \).
The angle is equal to 4 of the acute angles of the pattern block.
So, the measure of the angle is \( 120^{\circ} \).
Example 2:
Find the measure of \( \angle POQ \) using a protractor. Then classify it.
Solution:
Step 1: Place the protractor over the angle such that the protractor’s center coincides with the angle’s vertex O.
Step 2: Align one side of the angle, \( \overset{\rightarrow}{OQ} \), with the \( 0^{\circ} \) mark on the inner scale of the protractor.
Step 3: Observe the mark on the protractor where the other side of the angle, \( \overset{\rightarrow}{OP} \), passes through the inner scale.
So, the measure of \( \angle POQ \) is \( 60^{\circ} \) . It is an acute angle.
Example 3:
Draw \( \angle POQ \) that measures \( 70^{\circ} \).
Solution:
Step 1: Place the center of the protractor on point O. Align \( \overset{\rightarrow}{OQ} \) with the \( 0^{\circ} \) mark on the inner scale of the protractor.
Step 2: Using the same scale mark a point at \( 70^{\circ} \). Label the point P.
Step 3: Use the protractor to draw \( \overset{\rightarrow}{OP} \).
Example 4:
A contractor constructs two roofs, roof 1 and roof 2. How much greater is the angle measure of roof 1 than that of roof 2?
Solution:
Measure the angle of each roof with a protractor.
The angle measure of roof 1 is \( 45^{\circ} \) and the angle measure of roof 2 is \( 30^{\circ} \).
Subtract the angle measure of roof 2 from the angle measure of roof 1.
\( 45^{\circ}-35^{\circ}=15^{\circ} \)
Hence, the angle measure of roof 1 is \( 15^{\circ} \) greater than the angle measure of roof 2.
Example 5:
Use the given pattern block to find the measure angle made by the hand of the clock.
Solution:
Each angle of the given pattern block has a measure of \( 90^{\circ} \). The angle made by the clock’s hands is equal to 2 of the angles of the pattern block.
So, the measure of the angle formed by the clock’s hands is \( 120^{\circ} \).
When two rays intersect at a point, an angle is formed. An “angle” is the “opening” between these two rays, which is represented by the symbol “∠”. Angles are usually expressed in degrees, denoted by “\( ^{\circ}\)”, such as \( 60^{\circ},~90^{\circ}\), and so on.
Acute and obtuse angles are the two types of angles that are less than 180 degrees. Acute angles are always less than 90 degrees, while obtuse angles are more than 90 degrees but always less than 180 degrees.
A straight angle is equal to 180 degrees. A straight line always forms a straight angle.