Question

12 mark for review
in triangle xyz, angle y is a right angle, point p lies on xz, and point q lies on yz such that pq is parallel to xy. if the measure of angle xzy is 63°, what is the measure, in degrees, of angle xpq?

Explanation:

Step1: Analyze triangle XYZ

In right triangle \( XYZ \), \( \angle Y = 90^\circ \), \( \angle XZY = 63^\circ \). So \( \angle XYZ = 90^\circ \), \( \angle ZXY = 180^\circ - 90^\circ - 63^\circ = 27^\circ \)? Wait, no, wait. Wait, \( PQ \parallel XY \), so let's think about the lines.

Step2: Use parallel lines and transversal

Since \( PQ \parallel XY \), and \( XZ \) is a transversal. Wait, but also, \( \angle XZY = 63^\circ \), and \( PQ \parallel XY \), so \( \angle PQZ = \angle XYZ = 90^\circ \) (corresponding angles). Then in triangle \( PQZ \), \( \angle PZQ = 63^\circ \), \( \angle PQZ = 90^\circ \), so \( \angle QPZ = 27^\circ \)? Wait, no, we need \( \angle XPQ \). Wait, \( \angle XPQ + \angle QPZ = 180^\circ \) (linear pair). Wait, no, maybe another approach.

Wait, \( XY \parallel PQ \), so \( \angle XYZ = \angle PQZ = 90^\circ \) (corresponding angles). In triangle \( XYZ \), \( \angle XZY = 63^\circ \), so \( \angle ZXY = 180^\circ - 90^\circ - 63^\circ = 27^\circ \). Now, since \( XY \parallel PQ \), \( \angle ZXY + \angle XPQ = 180^\circ \)? No, wait, maybe alternate interior angles? Wait, no, \( XZ \) is a transversal cutting \( XY \) and \( PQ \). Wait, \( XY \) and \( PQ \) are parallel, so \( \angle XYZ \) and \( \angle PQZ \) are corresponding angles (both right angles). Then, \( \angle XZY = 63^\circ \), so in triangle \( PQZ \), \( \angle QPZ = 180^\circ - 90^\circ - 63^\circ = 27^\circ \). Then, \( \angle XPQ \) and \( \angle QPZ \) are supplementary (since they form a linear pair on line \( XZ \)), so \( \angle XPQ = 180^\circ - 27^\circ = 153^\circ \)? Wait, no, that can't be. Wait, maybe I messed up the direction.

Wait, let's draw the triangle. Right angle at \( Y \), \( XZ \) is hypotenuse. \( P \) is on \( XZ \), \( Q \) is on \( YZ \), \( PQ \parallel XY \). So \( XY \) is vertical (let's say), \( YZ \) is horizontal, right angle at \( Y \). Then \( PQ \parallel XY \), so \( PQ \) is also vertical? No, \( XY \) is from \( X \) to \( Y \) (vertical), \( YZ \) is from \( Y \) to \( Z \) (horizontal). Then \( PQ \) is parallel to \( XY \), so \( PQ \) is vertical, so \( Q \) is on \( YZ \), so \( PQ \) is vertical, so \( PQ \perp YZ \), same as \( XY \perp YZ \). Then \( \angle XZY = 63^\circ \), so in triangle \( XYZ \), \( \angle ZXY = 27^\circ \). Now, \( XZ \) is the hypotenuse, \( P \) is on \( XZ \), \( PQ \) is vertical, so \( \angle XPQ \): let's see, \( XY \) and \( PQ \) are both vertical, so \( X \) to \( Y \) to \( Q \) to \( P \) to \( X \)? Wait, maybe \( \angle XPQ \) is equal to \( 180^\circ - \angle ZXY \)? Wait, no, let's use the fact that \( PQ \parallel XY \), so \( \angle XYZ + \angle PQZ = 180^\circ \)? No, they are both 90, so that's not. Wait, maybe the sum of angles in a triangle. Wait, \( \angle XZY = 63^\circ \), \( \angle PQZ = 90^\circ \), so \( \angle QPZ = 27^\circ \), then \( \angle XPQ = 180^\circ - 27^\circ = 153^\circ \). Wait, that makes sense because \( P \) is on \( XZ \), so \( \angle XPQ \) and \( \angle QPZ \) are adjacent and form a straight line (linear pair), so they add up to 180. So \( \angle QPZ = 27^\circ \), so \( \angle XPQ = 180 - 27 = 153 \)? Wait, but let's check again.

In triangle \( XYZ \), right-angled at \( Y \), so \( \angle X + \angle Z = 90^\circ \). Wait, \( \angle Z = 63^\circ \), so \( \angle X = 27^\circ \). Now, \( PQ \parallel XY \), so \( \angle X + \angle XPQ = 180^\circ \) (consecutive interior angles, since \( XZ \) is a transversal). So \( 27^\circ + \angle XPQ = 180^\circ \), so \( \angle XPQ = 153^\circ \). Yes, that's correct. Consecutive interior angles are supplementary when lines are parallel. So \( XY \parallel PQ \), \( XZ \) is transversal, so \( \angle X \) and \( \angle XPQ \) are consecutive interior angles, so they add to 180. So \( \angle XPQ = 180 - 27 = 153 \).

Answer:

\( 153 \)