two circles intersect each other at point a and b .if ap and aq are diameters prove that pbq is a straight line

Dear Student,

Consider the figure as given below:



Given: AP and AQ are diameters.

To Prove: PBQ is a straight line.

Construction: Join AB

Proof: AP is a diameter $\Rightarrow \angle ABP=90° \left[Angle in a semicircle\right]$
AQ is a diameter $\Rightarrow \angle ABQ=90° \left[Angle in a semicircle\right]$
$$\begin{gathered} \angle PBQ=\angle ABP+\angle ABQ \\ =90°+90° \\ =180° \end{gathered}$$

Thus, PBQ is a straight line.
Hence Proved
Regards,