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

#### Options

(i) only

(ii) only

(iii) only

(ii) and (iii) only

#### Solution

Now, let us consider each statement one by one:

(i) Statement: a+b+ = d+ c

This statement is __incorrect__

__Explanation__:

We have, *a* and *d* are vertically opposite angles.

Therefore,

a = d (I)

Similarly, *b* and *e* are vertically opposite angles.

Therefore,

b =e (II)

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

a + b = d + e

Thus, this statement is incorrect.

(ii) Statement: a+ c+ e = 180°

This statement is __correct.__

__Explanation__:

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

a +f +e = 180° (III)

Also f°and c°are vertically opposite angles, therefore, these must be equal.

f = c

Putting f = c in (III), we get:

a + c + e = 180°

(iii) Statement: b + f = a + e

This statement is correct.’

__Explanation:__

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

a + f + b = 180° (IV)

__A__lso c°, d° and e° form a linear pair, therefore their sum must be supplementary.

c + d + e = 180° (V)

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

a+ f +b = c + d + e

Also a°and d°are vertically opposite angles, therefore, these must be equal.

Therefore,

a = b

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

a + f + b = c + d + e

a + f + b = c+ a + e

b + f = a + e

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