Find the value of the angles marked with letters in the following diagrams (a) and (b).

The two horizontal lines are parallel, as indicated by the arrows. The left diagonal line is a transversal cutting these parallel lines.
The angle marked $45^\circ$ and the angle marked $x$ are alternate interior angles.
Therefore, they are equal.

$$x = 45^\circ$$

The angle $126^\circ$ and the interior angle of the triangle on the bottom right form a linear pair (angles on a straight line).
Let this interior angle be $\alpha$.

$$\alpha + 126^\circ = 180^\circ$$ $$\alpha = 180^\circ - 126^\circ$$ $$\alpha = 54^\circ$$

Consider the triangle formed by the two diagonal lines and the bottom parallel line. The angles inside this triangle are $x$, $y$, and $\alpha$.
The sum of angles in a triangle is $180^\circ$.

$$x + y + \alpha = 180^\circ$$

Substitute the values of $x$ and $\alpha$:

$$45^\circ + y + 54^\circ = 180^\circ$$ $$99^\circ + y = 180^\circ$$ $$y = 180^\circ - 99^\circ$$ $$y = 81^\circ$$

The final answers are:

• $x = \boxed{45^\circ}$
• $y = \boxed{81^\circ}$