In a quadrilateral ^@PQRS,^@ if ^@PQ \parallel RS, \angle S = 2 \angle Q, PS = q^@ and ^@RS = p.^@ Find the length of the side ^@PQ.^@


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Answer:

^@ p + q ^@

Step by Step Explanation:
  1. Let us first draw the quadrilateral ^@PQRS^@.
      P Q R S 2x x E
    Let ^@ \angle Q = x ^@
    So, ^@ \angle S = 2x ^@
    Let us join ^@R^@ to a point ^@E^@ on the side ^@PQ^@ such that ^@PERS^@ is a parallelogram.
  2. We know that opposite sides of a parallelogram are equal.
    So, ^@ \angle PSR = \angle PER = 2x \ldots (i) ^@
  3. Also, @^ \begin{align} & \angle PER + \angle REQ = 180^\circ &&[ \text{ Angles on a straight line } ] \\ \implies & \angle REQ = 180^ \circ - \angle PER \\ \implies & \angle REQ = 180^ \circ - 2x && [\text{From } (i)] \\ \end{align} @^
  4. The sum of angles of a triangle is ^@ 180^ \circ. ^@
    In ^@ \triangle ERQ^@ @^ \begin{align} & \angle REQ + \angle EQR + \angle QRE = 180^ \circ \\ \implies & 180^\circ - 2x + x + \angle QRE = 180^\circ \\ \implies & \angle QRE = x \end{align} @^
  5. In ^@ \triangle ERQ, ^@ @^ \begin{align} & \angle QRE = \angle EQR \\ \implies & ER = EQ \ldots (ii) && [\text{ Sides opposite to equal angles are equal. }] \\ \end{align} @^
  6. We are given that ^@ RS = p ^@ and ^@ PS = q. ^@
    As, opposite sides of a parallelogram are equal,
    ^@ RS = PE = p ^@
    and ^@ PS = ER = q ^@
    ^@ \implies EQ = q \ldots [\text{From } (ii)] ^@
  7. We can see that ^@ PQ = PE + EQ = p + q ^@
  8. Thus, ^@ PQ = p + q ^@

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