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

Step 1: Find the value of $x$. 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. $x = 45^\circ$

Step 2: Find the interior angle adjacent to $126^\circ$. 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 $\alpha$. $\alpha + 126^\circ = 180^\circ$ $\alpha = 180^\circ - 126^\circ$ $\alpha = 54^\circ$

Step 3: Find the value of $y$. 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:

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