1. **Problem statement:** We need to find the size of angle $x$ given two parallel horizontal lines crossed by two diagonal transversal lines, with one angle marked $45^\circ$ adjacent to angle $x$. 2. **Angle facts used:** - Corresponding angles between parallel lines are equal. - Angles on a straight line sum to $180^\circ$. 3. **Step-by-step solution:** - The pink angle is $45^\circ$. - Angle $x$ is adjacent to the $45^\circ$ angle on the same straight line, so their sum is $180^\circ$. 4. **Calculate $x$:** $$ x + 45^\circ = 180^\circ $$ $$ x = 180^\circ - 45^\circ $$ $$ x = 135^\circ $$ 5. **Final answer:** Angle $x$ is $135^\circ$. This uses the fact that angles on a straight line add up to $180^\circ$.