In a quadrilateral PQRS, if PQ∥RS,∠S=2∠Q,PS=q and RS=p. Find the length of the side PQ.
Answer:
p+q
- Let us first draw the quadrilateral PQRS.
So, ∠S=2x
Let us join R to a point E on the side PQ such that PERS is a parallelogram. - We know that opposite sides of a parallelogram are equal.
So, ∠PSR=∠PER=2x…(i) - Also, ∠PER+∠REQ=180∘[ Angles on a straight line ]⟹∠REQ=180∘−∠PER⟹∠REQ=180∘−2x[From (i)]
- The sum of angles of a triangle is 180∘.
In △ERQ ∠REQ+∠EQR+∠QRE=180∘⟹180∘−2x+x+∠QRE=180∘⟹∠QRE=x - In △ERQ, ∠QRE=∠EQR⟹ER=EQ…(ii)[ Sides opposite to equal angles are equal. ]
- We are given that RS=p and PS=q.
As, opposite sides of a parallelogram are equal,
RS=PE=p
and PS=ER=q
⟹EQ=q…[From (ii)] - We can see that PQ=PE+EQ=p+q
- Thus, PQ=p+q