In the given figure, which of the following statements must be true?
(i) a + b = d + c

(ii) a + c + e = 180°

(iii) b + f = c + e

(a) (i) only

(b) (ii) only

(c) (iii) only

(d) (ii) and (iii) only

Now, let us consider each statement one by one:

(i)

Statement:

This statement is incorrect

Explanation:

We have, a and d are vertically opposite angles.

Therefore,

(I)

Similarly, b and e are vertically opposite angles.

Therefore,

(II)

On adding (I) and (II), we get:

Thus, this statement is incorrect.

(ii)

Statement:

This statement is correct.

Explanation:

As , and form a linear pair, therefore their sum must be supplementary.

(III)

Also andare vertically opposite angles, therefore, these must be equal.

Putting in (III), we get:

(iii)

Statement:

This statement is correct.’

Explanation:

As, and form a linear pair, therefore their sum must be supplementary.

(IV)

Also , and form a linear pair, therefore their sum must be supplementary.

(V)

On comparing (IV) and (V), we get:

Also andare vertically opposite angles, therefore, these must be equal.

Therefore,

Substituting the above equation in (VI), we get:

Thus, out of all, (ii) and (iii) are correct.

Hence, the correct choice is (d).