Question

If the sum of three angles (in degrees) of a quadrilateral is a perfect square number but the fourth angle is acute, then the sum of all possible measures of the fourth angle is

• A. ${107}^0$
• B. ${118}^0$
• C. ${143}^0$
• D. ${180}^0$

Answer 1

The correct option is A ${107}^0$

The possible perfect square values of the sum of three angles of a quadrilateral such that the fourth angle is an acute angle, are $324°$ and $289°$, i.e., $(18{)}^2$ and $(17{)}^2$ respectively.
If the sum is $324°$, then the fourth angle is $(360°–324°)=36°$ whereas if the sum is $289°$, then the fourth angle is $(360°–289°)=71°.$
Therefore, the sum of all possible measures of the fourth angle $=36°+71°=107°$
Hence, the correct answer is option (a).