Question

AB, CD and EF are three lines intersecting at the same point.
(i) Find x, if y = 45° and z = 90°.
(ii) Find a, if x = 3a, y = 5x and r = 6x.

Answer 1

AB, CD and EF are intersecting each other at O.
and ∠DOF = x°, ∠AOC = y°
and ∠BOE = z°
But ∠DOB = ∠AOC = y° ...............(Vertically opposite angles)

Similarly, ∠COE = ∠DOF = x°
and ∠AOF = ∠BOE = z°

∴ CD is a straight line

∴ ∠COE + ∠BOE + ∠DOE = 180°

⇒ x° + z° + y° = 180°

⇒ x° + y° + z° = 180°

(i) If y = 45°, and z = 90°, then

⇒ x° + 45° + 90° = 180°

⇒ x° + 135° = 180°

∴ x° = 180°− 135° = 45°

(ii) If x = 3a, y = 5x, z = 6x,
then x + y + z = 180°

⇒ x + 5x + 6x = 180°

⇒ 12x = 180°

⇒ x = ${\displaystyle \frac{180°}{12}}$ = 15°

But x = 3a

∴ 3a = 15°

⇒ a = ${\displaystyle \frac{15°}{3}}$ = 5°

Hence a = 5°