In each of the following figures, AB || CD and EF is a transversal. Find each one of the unknown angles x, y, z in each case.

(i)

(ii)

(iii)

Given:

AB || CD

EF is a transversal

(i)

x° = 55° $\quad$[Vertically opposite angles are equal]

y° and 55° angle are alternate interior angles i.e., these angles are on opposite sides of the transversal between the parallel lines. So, they are equal.

∴ y° = 55°

Since y° and z° form a linear pair on line CD they must sum to 180°.

∴ y° + z° = 180°

⇒ 55° + z° = 180° $\quad$[Substituting the value of y]

⇒ z° = 180° - 55°

⇒ z° = 125°

x° = 55°, y° = 55°, z° = 125°

(ii)

x° and 130° are corresponding angles i.e., these angles are in the same relative position at each intersection. So, they are equal.

∴ x° = 130°

Since x° and y° form a linear pair on line CD they must sum to 180°.

∴ x° + y° = 180°

⇒ 130° + y° = 180° $\quad$[Substituting the value of x]

⇒ y° = 180° - 130°

⇒ y° = 50°

z° = y° $\quad$[Corresponding angles]

∴ z° = 50°

x° = 130°, y° = 50°, z° = 50°

(iii)

From the figure,

z° = 40° $\quad$[Vertically opposite angles]

z° = y° $\quad$[Interior alternate angles]

∴ y° = 40°

Since x° and y° form a linear pair on line AB they must sum to 180°.

∴ x° + y° = 180°

⇒ x° + 40° = 180° $\quad$[Substituting the value of y]

⇒ x° = 180° - 40°

⇒ x° = 140°

x° = 140°, y° = 40°, z° = 40°