Question

The given figure shows a circle with centre O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate:

1. angle QTR
2. angle QRP
3. angle QRS
4. angle STR

Answer 1

Join PQ, RQ and ST.

i. ∠POQ + ∠QOR = 180°

`=>` 100° + ∠QOR = 180°

`=>` ∠QOR = 80°

Arc RQ subtends ∠QOR at the centre and ∠QTR at the remaining part of the circle.

∴ `∠QTR = 1/2 ∠QOR`

`=> ∠QTR = 1/2 xx 80^circ = 40^circ`

ii. Arc QP subtends ∠QOP at the centre and ∠QRP at the remaining part of the circle.

∴  `∠QRP = 1/2 ∠QOP`

`=> ∠QRP = 1/2 xx 100^circ = 50^circ`

iii. RS || QT

∴ ∠SRT = ∠QTR  ...(Alternate angles)

But ∠QTR = 40°

∴ ∠SRT = 40°

Now,

∠QRS = ∠QRP + ∠PRT + ∠SRT

`=>` ∠QRS = 50° + 20° + 40° = 110°

iv. Since RSTQ is a cyclic quadrilateral

∴ ∠QRS + ∠QTS = 180°   ...(Sum of opposite angles)

`=>` ∠QRS + ∠QTS + ∠STR = 180°

`=>` 110° + 40° + ∠STR = 180°

`=>` ∠STR = 30°