ABCD is a cyclic quadrilateral. AB and DC are produced to meet at P. AD and BC are produced to meet at Q. If $\angle AQB=2x^\circ$, $\angle QAB=3x^\circ$ and $\angle BPC=x^\circ$, then what is the value of $x$?
Solution
✅ Correct Option: 4
Let the minor arcs $AB,BC,CD,DA$ be $a,b,c,d$ degrees respectively. Using the inscribed-angle theorem and the external-secant angle theorem: | Given angle | Arc relation | |---|---| | $\angle QAB=3x$ | $b+c=6x$ | | $\angle AQB=2x$ | $a-c=4x$ | | $\angle BPC=x$ | $d-b=2x$ | From the relations, $a=c+4x$ and $d=b+2x$. The four arcs total $360^\circ$: $a+b+c+d=(c+4x)+b+c+(b+2x)=2(b+c)+6x$ Using $b+c=6x$: $360=12x+6x=18x$ $x=20$.