$PQRS$ is a cyclic quadrilateral. If $\angle P$ is 4 times $\angle R $, and $\angle S $ is 3 times $\angle Q$, then the average of $\angle Q$ and $\angle R $ is:
Answer & explanation
Correct answer: option 1
According to question
= \(\angle\)P = 4 x \(\angle\)R
= \(\angle\)P : \(\angle\)R = 4 : 1
By using the property
= \(\angle\)P + \(\angle\)R = 180
= 4x + x = 180
= 5x = 180
= x = 36
= \(\angle\)R = \({36}^\circ\)
Now,
= \(\angle\)S = 3 x \(\angle\)Q
= \(\angle\)S : \(\angle\)Q = 3 : 1
= 3y + y = 180
= 4y = 180
= y = 45
= \(\angle\)Q = \({45}^\circ\)
Average of \(\angle\)R and \(\angle\)Q is
= (\(\angle\)R + \(\angle\)Q)/2
= \(\frac{36\;+\;45}{2}\)
= \(\frac{81}{2}\) = \({40.5}^\circ\)
Therefore, the average of (\(\angle\)R and \(\angle\)Q) is \({40.5}^\circ\).