Question

The angles of a quadrilateral are \[\left( {x + 10} \right),\left( {2x + 5} \right),\left( {2x - 20} \right)\& \left( {2x - 5} \right)\], then find their measures.

Answer 1

Hint: First of all, find the sum of the angles in the quadrilateral and equate their sum to \[{360^0}\]. Then we will get the value of the variable \[x\]. Substitute the value of \[x\] in the given angles to get the required answer.

Complete Step-by-step answer:
Let the quadrilateral be ABCD.
The given angles of the quadrilateral are \[\left( {x + 10} \right),\left( {2x + 5} \right),\left( {2x - 20} \right)\& \left( {2x - 5} \right)\] as shown in the below figure:

We know that the sum of the angles in a quadrilateral is equal to \[{360^0}\].
So, we have
\[
   \Rightarrow x + 10 + 2x + 5 + 2x - 20 + 2x - 5 = 360 \\
   \Rightarrow x + 2x + 2x + 2x + 10 + 5 - 20 - 5 = 360 \\
   \Rightarrow 7x - 10 = 360 \\
   \Rightarrow 7x = 360 + 10 = 370 \\
  \therefore x = \dfrac{{370}}{7} = 52.86 \\
\]
Hence the measures of the angles are given by
\[
   \Rightarrow x + 10 = \dfrac{{370}}{7} + 10 = 52.86 + 10 = 62.86 \\
   \Rightarrow 2x + 5 = 2\left( {\dfrac{{370}}{7}} \right) + 5 = 105.71 + 5 = 110.71 \\
   \Rightarrow 2x - 20 = 2\left( {\dfrac{{370}}{7}} \right) - 20 = 105.71 - 20 = 85.71 \\
   \Rightarrow 2x - 5 = 2\left( {\dfrac{{370}}{7}} \right) - 5 = 100.72 \\
\]
Thus, the angles of the quadrilateral are \[62.86,110.71,85.71\& 100.72\].

Note: The sum of the angles in a quadrilateral is equal to \[{360^0}\]. For the angles of the quadrilateral we have rounded up the values to two decimal points. To verify the answer, add up the obtained values of angles of the quadrilateral. If their sum is equal to \[{360^0}\], then our answer is correct otherwise wrong.