Question

How many acute angles does a quadrilateral can have?

Answer 1

Hint: Acute angle means the angle whose value is less than ninety degree, for any figure if the angle is given between the sides, and then you can comment on its property of acute or obtuse. In a quadrilateral you should know that the sum of all interior angles are “360” and using this you can find the number of acute angles in a quadrilateral.

Complete step by step solution:
 According to the given question let’s see the cases which we can found as per requirement of the question,
Case 1: Let us assume that all the angles of quadrilateral are acute:
Now the maximum acute angle is the angle just before ninety, let us assume a variable “x” for that angle, now on adding with conditions we get:
\[
\Rightarrow x < 90 \\
\Rightarrow x + x + x + x < 90 + 90 + 90 + 90 \\
\Rightarrow x + x + x + x < 360 \\
\]
Here we have taken all four angles as just before the ninety degree and the result proves that four acute angles are not possible.
Case 2: Let us assume that three angles are acute:
Here we are assuming three angles as we previously assumed for the first case and the fourth angle is assumed as “y” which can be any angle greater than ninety degree, on solving we get:
\[
\Rightarrow x < 90\,and\,y > 90 \\
\Rightarrow x + x + x < 90 + 90 + 90\,and\,y > 90 \\
\Rightarrow 3x < 270\,and\,y > 90 \\
\Rightarrow combining\,both\,we\,get\,3x + y > 360\,\sin
ce\,y\,is\,greater\,then\,90\,and\,3x\,is\,just\,less\,then\,270 \\
\]
Hence maximum three acute angles are possible.

Note: The above question can be also solved just by saying that only four equal angles are possible to make a quadrilateral, hence any angle below it cannot make a quadrilateral, which proves maximum only three acute angles are possible to make a quadrilateral.