A right angle is that angle in geometry that measures 90˚. We cannot imagine our lives without right angles. A right angle is present everywhere. The top of your table and the legs are joined together at right angles. The floor of your house makes right angles to all the walls connected to it. When arranged properly, the pile of books makes right angles, and this list never ends. Therefore in this article, we will learn the fundamental concepts of right angles and how to make right angles and right-angled triangles. But first,
Do you all know what an angle is? If two rays coming from different directions intersect at a point, they produce an angle. The point of intersection is known as the vertex. Depending on the orientation of the rays, three major types of angles can be made they are listed below:
Acute angle: The angle formed by two rays less than 90˚ is termed an acute angle. Acute angle ranges from 0˚ to 90˚ (excluding 90˚).
Obtuse angle: The angle made by two rays having a value greater than 90˚ is called an obtuse angle. Obtuse angle ranges from 90˚ to 180˚ (excluding 90˚)
Right angle: When the rays form exactly an angle measuring 90˚, it is known as a right angle.
Note: Angles are measured in degrees with the symbol ‘ ˚ ’ or in radians with the abbreviation ‘rad’. In radians, the measure of a right angle is π/2. This is because in the radian metric system, the value of π = 180˚. Therefore 180˚/2 or π/2 = 90˚.
Now that we have covered all the basics let us dive deeper into the theories and concepts of right angles.
Where are Right Angles Present?
We have already discussed a few practical scenarios to find right angles. But that is not it! In this section, we shall learn about all the places to find right angles. Below is a list consisting of all the mathematical aspects of right angles:
- A right-angled triangle has one of its angles measuring 90˚.
- All the four sides of a rectangle and a square make right angles with each other, or we can say that all the four angles inside a rectangle and a square are 90˚.
- The diagonals of a square and a rhombus intersect at right angles.
- The corners of your room where you sleep or study are at right angles.
- A cube and cuboid have all the angles measuring 90˚. For example, a Rubik’s cube.
How to Make a Right Angle?
The simplest way to make a right angle is by making two straight lines. Make one line horizontal and the other line vertical. Make them intersect each other. The angle formed between the two lines will be the right angle. Make sure to keep the lines vertically and horizontally straight. You can use a scale for this purpose. But this method is not always accurate. Many times your hand may slip, resulting in a distorted line. Let us learn two ways to always make the perfect right angle.
Method 1: Using Protractor
Step 1: Grab a scale and draw a straight horizontal line in your notebook.
Step 2: Now, take your protractor and place it above the horizontal line. This should be done in such a way that the last bottom-most line of the protractor and the horizontal line you made coincides.
Step 3: Look at your protractor and make a point just above the point written as 90. This is the 90˚ point from the horizontal line.
Step 4: Again, grab the scale and make a straight line using the protractor to the horizontal line from the point you marked.
Step 5: You have made the perfect right angle.
Note: Always clean your rulers and protractors before working with them. They may contain graphite or ink from the last task they were used for and untidy your work.
Method 2: Using Compass
Step 1: Draw a horizontal line on your notebook, say AB.
Step 2: Grab your compass. Place your pencil in it and level the tip of the pencil with the tip of the compass. Now take an angle of any measure on it.
Step 3: Place the tip of the compass on point ‘A’ and move the pencil making an arc on the line.
Step 4: Mark the point where the arc touches the line ‘AB’ as ‘C’.
Step 5: Now place the tip at point ‘C’ and make another arc on the first arc. Make it a point ‘D’.
Step 6: Now place the tip at point ‘D’ and draw another arc on the first one. Make it a point ‘E’. Also, draw another arc using this point just above the horizontal line.
Step 7: Place the tip at point E and cut the arc you made above the horizontal line. Name the intersection as F.
Step 8: Draw a straight line going from point F to point A.
Step 9: You have successfully drawn a right angle.
You are now well versed with how to draw a perfect right angle. You can also try squares and set squares to make flawless right angles. They are like a scale but in the shape of a right-angled triangle. In the next section, we shall learn more about the right-angle triangle and how to find the angle of a right triangle.
Right Angle Triangle
When the value of one of the angles inside a triangle is exactly equal to 90 degrees, it is termed a right-angled triangle, as we all know that a triangle has three angles inside it. In a right-angled triangle, the rest of the two angles are always acute angles. The sum of the two acute angles must always be 90˚. This can be proved from the example below:
Let triangle ABC be a right-angled triangle having a right angle at A. We know that the angle sum property of a triangle states that all the three angles sum inside a triangle is equal to 180˚.
Therefore, A + B + C = 180˚.
90˚ + B + C = 180˚
B + C = 180˚ – 90˚
B + C = 90˚
Also based on parameters such as side and angles, right-angled triangles are of two types:
Isosceles Right Triangle: An isosceles right-angled triangle combines the properties of an isosceles triangle and a right-angled triangle. It is a triangle having two equal sides and the angle subtended by these lines is exactly equal to 90-degrees.
Right Scalene Triangle: A right scalene triangle combines the properties of a right-angled triangle and a scalene triangle. All the sides are unequal, and one of the angles measures 90 degrees in such types of triangles.
Let us now learn more terms related to a right-angled triangle:
Perpendicular: The height of the right angle triangle is known as the perpendicular.
Base: The side on which the right-angled triangle is made is the base.
Hypotenuse: The longest side of the right-angled triangle is termed the hypotenuse.
The perpendicular and the triangle base always form 90˚ angles with each other. Never will the hypotenuse and base or the hypotenuse and perpendicular make the right angle. They will only form the acute angles.
A right triangle is so special that Greek mathematician Pythagoras invented the formula for this triangle. The Pythagoras rule states that in every right-angled triangle, the square of the hypotenuse is equal to the sum of the square of perpendicular and base.
Formula: (Hypotenuse)2 = (Perpendicular)2 + (Base)2.
Some real-life examples of a right triangle are mentioned below:
- The slide made for children in the park is an example of a right-angled triangle.
- The best and the most popular example of a right-angled triangle is trigonometry. It is an important result that a whole branch of mathematics is based on trigonometry. Trigonometry is also the most feared mathematical topic in schools.
Important points to Keep in Mind:
- When two lines meet each other at right angles, the two lines are perpendicular to each other.
- A right-angle is 90 degrees, made with the base and the hypotenuse.
- Right angles are mostly used at construction sites to provide adequate support at the corners.
- They are used to support long-spanned bridges and culverts.
- Right angles are quite strong in practical life. The angle does not rupture upon heavy usage conditions. This is because it gets support from two sides.