Defining Perpendicular Lines
The definition of perpendicular lines states that when two lines meet or intersect each other, an angle of 90 degrees is formed. There are two images here, the first showing us the formation of a 90 degree angle when two lines meet at a point, and the second image shows us four 90 degree angles that are formed when two lines intersect each other at a point. Hence, when two lines form an angle of 90 degrees when they meet or even intersect each other they are known as perpendicular lines. Additionally, an angle that measures 90 degrees is also known as a right angle.
In this image we can see lines AB and BC meeting at the point B. The angle that is formed at point B is equal to 90 degrees. Hence, the measure of angle ABC is 90 degrees.
Or, ∠ABC = 90°
The second image shows us two intersecting lines that form four right angles. There are two lines, AB and CD intersecting at point O. It is at this point that we can observe the formation of four right angles, since AB and CD are perpendicular lines.
As stated earlier we can observe that the two lines in the image intersect each other and have formed these angles:
∠AOD = 90°
∠DOB = 90°
∠BOC = 90°
∠COA = 90°
How Do We Draw Perpendicular Lines?
There are different methods of drawing perpendicular lines. The simplest method is to use a ruler and a pencil to draw two straight lines that meet at a point, and these two lines resemble the shape of an ‘L’ forming a 90 degree angle.
By following the dotted lines we can draw two lines that are perpendicular lines.
Another method of drawing perpendicular lines is to draw two lines that resemble the ‘plus’ symbol to form perpendicular lines.
Even in this case, by following the dotted lines we can draw two lines that are perpendicular to each other and this results in the formation of four 90 degree angles.
Rapid Recall
Perpendicular lines AB and BC forming the angle, ∠ABC = 90°.
Perpendicular lines AB and CD intersect forming the angles,
∠AOD = 90°
∠DOB = 90°
∠BOC = 90°
∠COA = 90°
Solved Examples
Example 1:
According to the definition of perpendicular lines identify the perpendicular lines in the following images.
Image 1
Image 2
Image 3
Solution:
Image 1:
In the first image there are two right angles.
∠ADC = 90°
∠ADB = 90°
Hence, the two lines AD and BC are perpendicular lines.
Image 2:
In this image, the lines AB and CD do not form right angles at the point where they intersect.
So,
∠AOD ≠ 90°
∠DOB ≠ 90°
∠BOC ≠ 90°
∠COA ≠ 90°
Hence, the lines AB and CD are not perpendicular lines.
Image 3:
In the third image the lines AB and CD are intersecting at the point O. There are also four right angles measuring 90° that are formed.
∠AOD = 90°
∠BOD = 90°
∠COB = 90°
∠AOC = 90°
Hence, the line AB is perpendicular to line CD.
Example 2:
There are two traffic signs in the image. Identify whether the arrows on the symbols on each sign are perpendicular or not.
Solution:
The traffic sign to the left has two arrows intersecting each other, and there are right angles formed at the point of intersection.
Hence, the arrows are perpendicular.
The traffic sign to the right has two arrows intersecting each other but there are no right angles formed at the point of intersection.
Hence, the arrows are not perpendicular.
Example 3:
Julian has been given a puzzle by his teacher and it’s a jumble of lines intersecting each other at the points that are shown in the image. Can you help Julian find all the perpendicular lines?
Note that he needs to find just five sets of perpendicular lines.
Solution:
The image has the following perpendicular lines:
- VW and KL forming a right angle at point H
- MN and YZ forming a right angle at point I
- OP and KL forming a right angle at point P
- MI and IZ forming a right angle at point I
- NI and IJ forming a right angle at point I
Hence these are the perpendicular lines in the puzzle.
The main difference between parallel and perpendicular lines is that parallel lines do not meet and have a constant distance between them, but, in order for an angle to be formed two lines will have to meet as is the case for perpendicular lines which form a right angle.
Perpendicular lines are commonly seen at the edges of a book, the edges of a flat screen TV, the edges of a smartphone, the edges of a rectangular window and in many other instances.
It should be noted that perpendicular lines always form an angle of 90°.
A straight angle measures 180°, but on the other hand perpendicular lines always form a right angle measuring 90°.