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.
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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.

KnowledgeBoat · Accepted Answer

Join PB. In cyclic quadrilateral PBCQ, ⇒ ∠BPQ + ∠BCQ = 180° [Sum of opposite angles of a cyclic quadrilateral = 180°] ⇒ ∠BPQ + 140° = 180° ⇒ ∠BPQ = 180° - 140° ⇒ ∠BPQ = 40° ............(1) In △PBQ, ∠PBQ = 90° [Angle in a semi-circle is a right angle.] ⇒ ∠PBQ + ∠BPQ + ∠PQB = 180° [Angle sum property of triangle] ⇒ 90° + 40° + ∠PQB = 180° ⇒ 130° + ∠PQB = 180° ⇒ ∠PQB = 180° - 130° ⇒ ∠PQB = 50°. In cyclic quadrilateral PQBA, ⇒ ∠PQB + ∠PAB = 180° [Sum of opposite angles of a cyclic quadrilateral = 180°] ⇒ 50° + ∠PAB = 180° ⇒ ∠PAB = 180° - 50° ⇒ ∠PAB = 130°. In △PAB, ⇒ ∠PAB + ∠PBA + ∠BPA = 180° [Angle sum property of triangle] ⇒ 130° + ∠PBA + ∠BPA = 180° ⇒ ∠PBA + ∠BPA = 180° - 130° ⇒ ∠PBA + ∠BPA = 50° ................(2) Given, PA = PB Angles opposite to equal sides are equal. ∴ ∠PBA = ∠BPA = x (let) Substituting above value in (2), we get : ⇒ x + x = 50° ⇒ 2x = 50° ⇒ x = ⇒ x = 25°. From figure, ∠AQB = ∠APB = 25° [Angles in same segment are equal.] ∠APQ = ∠APB + ∠BPQ = 25° + 40° = 65. We know that, Angle subtended by an arc at the center is twice the angle subtended at any other point of circumference. Arc AQ subtends ∠AOQ at the center and ∠APQ at the remaining part of the circle. ∠AOQ = 2∠APQ = 2 × 65° = 130°. In △AOQ, OA = OQ [Radii of same circle] As, angles opposite to equal sides are equal. ∠OAQ = ∠OQA = y (let) ⇒ ∠OAQ + ∠OQA + ∠AOQ = 180° [Angle sum property of triangle] ⇒ y + y + 130° = 180° ⇒ 2y = 180° - 130° ⇒ 2y = 50° ⇒ y = ⇒ y = 25°. ∴ ∠OAQ = 25°. Since, ∠OAQ = ∠AQB = 25°. ∠OAQ and ∠AQB are alternate angles. Thus, AO and BQ are parallel. Hence, ∠AQB = 25° and ∠PAB = 130°.

Mathematics

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.

Circles

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Mathematics

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.

Circles

Related Questions

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In the given figure, AC = AE.
Show that :
(i) CP = EP
(ii) BP = DP

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
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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 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.