From the figure, find the values of $x$ and $y$.

1. Finding the value of $y$:

The angles at vertex $A$ lie on a straight line. Therefore, the interior angle $\angle A$ inside the triangle and the exterior angle $100^\circ$ form a linear pair (adding up to $180^\circ$):

$\angle A = 180^\circ - 100^\circ = 80^\circ$

Now, looking inside the triangle $\triangle ABC$, the sum of all interior angles must be $180^\circ$:

$\angle A + \angle B + y = 180^\circ$

$80^\circ + 50^\circ + y = 180^\circ$

$130^\circ + y = 180^\circ$

$y = 50$

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2. Finding the value of $x$:

The angles $x^\circ$ and $y^\circ$ lie on a straight horizontal line at vertex $C$, making them a linear pair:

$x + y = 180^\circ$

$x + 50^\circ = 180^\circ$

$x = 130$

(Alternatively, by the exterior angle theorem, $x$ is equal to the sum of the two opposite interior angles: $x = \angle A + \angle B = 80^\circ + 50^\circ = 130$.)

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Conclusion:

• $x = 130$
• $y = 50$