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( a ) $\dfrac{a}{2}+2b$

( a ) $a+2b$

( a ) $2a-b$

( a ) $a+b$

Answer
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In this question, we have given hint that ABCD is quadrilateral and AB is parallel to CD and angle D is twice of angle B that is $\angle D=2\times \angle B$ and side AD = b and side CD = a, then we have to evaluate the length of side AB.

Now, from figure we can see that AD = b and CM = b and CD = a and AM = a.

As, it is given in the question that, AB is parallel to CD, so we can say that ABCD is a parallelogram.

Also from the figure, we can say that $\angle D=\angle AMC$

Now, let angle B be angle x then as $\angle D=2\times \angle B$ so, angle D will be equals to 2x that is $\angle D=2x$or $\angle AMC=2x$

Now, from the figure in $\vartriangle MCB$,

$\angle AMC=\angle MCB+\angle x$

$2x=\angle MCB+\angle x$ by exterior angle sum property which says that exterior angle equals the sum of the opposite interior angle.

On solving, we get

$\angle MCB=x$

Also, $\angle MCB=\angle B$

So, $\angle MCB=\angle B=x$

In a triangle, if two angles are equal then sides opposite to equal angles are also equal.

So, in $\vartriangle MCB$

$MB = MC$

Now, from figure we can say that

$AB = AM + MB$, and $AM = A$ and $MB = b$, then

Or, $AB = a + b$