The given figure shows a circle with center O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate :
(i) angle QTR
(ii) angle QRP
(iii) angle QRS
(iv) angle STR

Question

The given figure shows a circle with center O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate :
(i) angle QTR
(ii) angle QRP
(iii) angle QRS
(iv) angle STR

KnowledgeBoat · Accepted Answer

(i) From figure, ⇒ ∠POQ + ∠QOR = 180° [Linear pairs] ⇒ 100° + ∠QOR = 180° ⇒ ∠QOR = 180° - 100° ⇒ ∠QOR = 80°. We know that, Angle subtended by an arc at the center is twice the angle subtended at any other point of circumference. Arc RQ subtends ∠QOR at the center and ∠QTR at the remaining part of the circle. ⇒ ∠QOR = 2∠QTR ⇒ ∠QTR = ∠QOR = Hence, ∠QTR = 40°. (ii) We know that, Angle subtended by an arc at the center is twice the angle subtended at any other point of circumference. Arc QP subtends ∠QOP at the center and ∠QRP at the remaining part of the circle. ⇒ ∠QOP = 2∠QRP ⇒ ∠QRP = ∠QOP = Hence, ∠QRP = 50°. (iii) Given, RS || QT ⇒ ∠SRT = ∠QTR = 40° (Alternate angles are equal) From figure, ∠QRS = ∠QRP + ∠PRT + ∠SRT = 50° + 20° + 40° = 110°. Hence, ∠QRS = 110°. (iv) Since, RSTQ is a cyclic quadrilateral and sum of opposite angles of cyclic quadrilateral = 180°. ⇒ ∠QRS + ∠QTS = 180° ⇒ ∠QRS + ∠QTR + ∠STR = 180° ⇒ 110° + 40° + ∠STR = 180° ⇒ ∠STR = 180° - 150° ⇒ ∠STR = 30°. Hence, ∠STR = 30°.

Mathematics

The given figure shows a circle with center O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate :
(i) angle QTR
(ii) angle QRP
(iii) angle QRS
(iv) angle STR

Circles

Related Questions

In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO
(ii) ∠AOB
(iii) ∠APB

The given figure shows a semi-circle with center O and diameter PQ. If PA = AB and ∠BCQ = 140°; find measures of angles PAB and AQB. Also, show that AO is parallel to BQ.

In the given figure, XY is the diameter of the circle and PQ is a tangent to the circle at Y.
If ∠AXB = 50° and ∠ABX = 70°, find ∠BAY and ∠APY.

In the given figure, QAP is the tangent at point A and PBD is a straight line.
If ∠ACB = 36° and ∠APB = 42°, find :
(i) ∠BAP
(ii) ∠ABD
(iii) ∠QAD
(iv) ∠BCD

Mathematics

The given figure shows a circle with center O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate :(i) angle QTR(ii) angle QRP(iii) angle QRS(iv) angle STR

Circles

Related Questions

In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find(i) ∠BCO(ii) ∠AOB(iii) ∠APB

The given figure shows a semi-circle with center O and diameter PQ. If PA = AB and ∠BCQ = 140°; find measures of angles PAB and AQB. Also, show that AO is parallel to BQ.

In the given figure, XY is the diameter of the circle and PQ is a tangent to the circle at Y.If ∠AXB = 50° and ∠ABX = 70°, find ∠BAY and ∠APY.

In the given figure, QAP is the tangent at point A and PBD is a straight line.If ∠ACB = 36° and ∠APB = 42°, find :(i) ∠BAP(ii) ∠ABD(iii) ∠QAD(iv) ∠BCD

The given figure shows a circle with center O such that chord RS is parallel to chord QT, angle PRT = 20° and angle POQ = 100°. Calculate :
(i) angle QTR
(ii) angle QRP
(iii) angle QRS
(iv) angle STR

In the adjoining figure, O is the centre of the circle. Tangents to the circle at A and B meet at C. If ∠ACO = 30°, find
(i) ∠BCO
(ii) ∠AOB
(iii) ∠APB

In the given figure, XY is the diameter of the circle and PQ is a tangent to the circle at Y.
If ∠AXB = 50° and ∠ABX = 70°, find ∠BAY and ∠APY.

In the given figure, QAP is the tangent at point A and PBD is a straight line.
If ∠ACB = 36° and ∠APB = 42°, find :
(i) ∠BAP
(ii) ∠ABD
(iii) ∠QAD
(iv) ∠BCD