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# Angle $\angle x$ and $\angle y$ are the exterior angles of a $\Delta ABC$, at the points $B$ and $C$. Also $\angle B > \angle C$, then relation between $\angle x$ and $\angle y$ isA) $\angle x > \angle y$B) $\angle x < \angle y$C) $\angle x = \angle y$D) None of these

Last updated date: 11th Sep 2024
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Hint:
In this question, first draw the triangle as per the given arrangement and then use the concept of the supplementary angles is used that is two angles are said to be supplementary angles if the sum of the angles are added up to $180^\circ$.

Complete step by step solution:
In this question, the exterior angles of the $\Delta ABC$ are $\angle x$ and $\angle y$ at point $B$ and $C$. One condition is given between the two angles of the triangle that is $\angle B > \angle C$, now the relation between the exterior angles is asked.
For the solution of the question, first draw a triangle $ABC$ in which angle $A$, angle $B$, and angle $C$ are the interior angle of the triangle. The angle $x$ and angle $y$ are the exterior angles of the triangle as shown in the below figure.

We know that the supplementary angles are the two angles whose sum will be $180^\circ$. From the above figure angle $x$ and angle $B$ are supplementary angle, so it can be written as,
$\angle B + \angle x = 180^\circ$
Now, we subtract the value of angle $x$ from both sides as,
$\angle B + \angle x - \angle x = 180^\circ - \angle x$
Solve the above expression and mark it as equation (1),
$\angle B = 180^\circ - \angle x$ (1)

Similarly, from the above figure angle $x$ and angle $B$ are supplementary angle, so it can be written as,
$\angle C + \angle y = 180^\circ$
Now, we subtract the value of angle $y$ from both sides as,
$\angle C + \angle y - \angle y = 180^\circ - \angle y$
Solve the above expression and mark it as equation (2),
$\angle C = 180^\circ - \angle y$ (2)
Now, the condition of angle $B$ and angle $C$ is given as,
$\angle B > \angle C$
Now, we substitute the values of angle $B$ and angle $C$ in the above equation as,
$\angle B > \angle C \\ 180^\circ - \angle x > 180^\circ - \angle y \\ 180^\circ - 180^\circ + \angle y > \angle x \\ \angle y > \angle x \\$

Therefore, the exterior angle $y$ is greater than angle $x$, so the correct option is (b).

Note:
Do not confuse the supplementary and the complementary angles. The supplementary angles are the two angles whose sum will be $180^\circ$ and the complementary angles are the two angles whose sum will be $90^\circ$.