Angle Y Measure Ea106F

1. **State the problem:** We are given two parallel lines $PQ$ and $AB$ cut by a transversal $PCB$. We know the angle at $P$ is $53^\circ$, the angle at $A$ is $37^\circ$, and we want to find the measure of angle $y$ at point $C$. 2. **Recall important rules:** When a transversal crosses parallel lines, alternate interior angles are equal, and the sum of angles on a straight line is $180^\circ$. 3. **Analyze the angles:** - Since $PQ \parallel AB$, angle at $P$ ($53^\circ$) and angle at $A$ ($37^\circ$) are on opposite sides of the transversal but not alternate interior angles directly. - The angle at $A$ ($37^\circ$) and angle $y$ at $C$ are adjacent angles on a straight line formed by the transversal and line $AB$. 4. **Calculate angle $y$:** - Angles on a straight line sum to $180^\circ$. - Therefore, $$y + 37^\circ = 180^\circ$$ - Solving for $y$: $$y = 180^\circ - 37^\circ = 143^\circ$$ 5. **Check for other angle relations:** - The angle at $P$ ($53^\circ$) and angle $y$ at $C$ are corresponding angles because $PQ \parallel AB$ and $PCB$ is the transversal. - Corresponding angles are equal, so $y = 53^\circ$. 6. **Resolve the contradiction:** - The angle adjacent to $37^\circ$ on line $AB$ is $143^\circ$, but $y$ is inside the lower angle at $C$, which corresponds to the $53^\circ$ angle at $P$. - Hence, the measure of $y$ is $53^\circ$. **Final answer:** $y = 53^\circ$ (option a).