In the adjoining figure, line a || line b. line l is a transversal. Find the measures of ∠x, ∠y, ∠z using the given information.
Let us mark the points A and B on l; K and M on a; L and N on b.
Suppose KM and LN intersect AB at P and Q respectively.
Since, a||b and l is a transversal, then
m∠PQL = m∠APK (Corresponding angles)
⇒ x = 105°
Since, AB and LN are straight lines that intersect at Q, then
m∠BQN = m∠PQL (Vertically opposite angles)
⇒ y = x
⇒ y = 105°
Since, AB is a straight line and ray QN stands on it, then
m∠BQN + m∠PQN = 180° (Angles in linear pair)
⇒ y + m∠PQN = 180°
⇒ 105° + m∠PQN = 180°
⇒ m∠PQN = 180° − 105°
⇒ m∠PQN = 75°
Now, m∠APM = m∠PQN (Corresponding angles)
⇒ z = 75°