question_answer

In the figure given, if \[AB||CD,\] what are the respective values of 'p' and 'q'?

A)  \[{{75}^{o}}\] and \[{{20}^{o}}\]  

B)  \[{{20}^{o}}\] and \[{{75}^{o}}\]

C)  \[{{25}^{o}}\] and \[{{70}^{o}}\] 

D)  \[{{70}^{o}}\] and \[{{25}^{o}}\]  

Correct Answer: D

Solution :

  The lines AB and EF intersect at G. \[\therefore \] \[\angle EGB=\angle AGF\] (Vertically opposite angles) \[\Rightarrow \] \[\angle AGF={{65}^{o}}\] Since  \[AB||CD,\] \[\angle GHD=\angle AGH=\angle AGF\]                 \[\Rightarrow \] \[\angle GHD={{65}^{o}}\]                 \[\Rightarrow \] \[\angle GHO+\angle OHD={{65}^{o}}\]                 \[\Rightarrow \] \[{{q}^{o}}={{65}^{o}}-{{40}^{o}}={{25}^{o}}\]                 Draw a line XY through ?O? parallel to AB and CD. Since \[XY||AB,\] \[\angle XOG=\angle BGO\] \[\Rightarrow \] \[\angle XOG={{45}^{o}}\] (Alternate angles) and \[XY||CD\,\,\,\Rightarrow \,\,\,\angle XOH=\angle OHD\] \[\Rightarrow \]  But                     &