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प्रश्न
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
पर्याय
(i) only
(ii) only
(iii) only
(ii) and (iii) only
उत्तर
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)
Also 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.
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