Question

Angles of a quadrilateral are in the ratio 1: 2: 3: 4. Find all angles. What special name can you give to this quadrilateral and why?

Answer 1

Hint: To solve this question we should know that the sum of all the interior angles of the quadrilateral is 360 degrees.

Complete step by step solution:
In the question, we have to find the angles of a quadrilateral that are in the ratio 1: 2: 3: 4.
Now, it is known that if the ratio is given as a : b : c : d, then the actual numbers are ax, bx, cx, dx, where x is the common factor of all the numbers.
So, now the angles that are in ratio of 1: 2: 3: 4, can be written as x, 2x, 3x and 4x in degrees.
Here the angles are in clockwise order. The figure can be as follows:

Now, it is very well known that the sum of all the interior angles of the quadrilateral is 360 degrees.
So, the angles x, 2x, 3x and 4x in degrees are add as follows:
\[\begin{align}
  & \Rightarrow x+2x+3x+4x={{360}^{\circ }} \\
 & \Rightarrow 10x={{360}^{\circ }} \\
 & \Rightarrow x={{36}^{\circ }} \\
\end{align}\]
So here the angles will be \[x={{36}^{\circ }}\], \[2x={{72}^{\circ }}\], \[3x={{108}^{\circ }}\], and \[4x={{144}^{\circ }}\].

Now we can see that \[2x+3x={{72}^{\circ }}+{{108}^{\circ }}={{180}^{\circ }}\] and \[x+4x={{36}^{\circ }}+{{144}^{\circ }}={{180}^{\circ }}\]
So the adjacent pair of angles is 180 degrees. So this is a trapezium as the sum of the adjacent angles on the parallel sides is 180 degrees.

Note: Trapezium is a kind of quadrilateral. The quadrilateral is a convex polygon where all interior angles are less than 180 degrees. It is important that we don’t get confused with the adjacent angles and the opposite angles. If the opposite pair of angles are the same for any quadrilateral, then it is a parallelogram and not trapezium.