Q) In the given figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 300. A chord RS is drawn parallel to tangent PQ. Find the ∠RQS.

Ans: In △PRQ, PQ and PR are tangents from an external point P to circle.

∴ PR = PQ

Since the angles opposites to equal sides are equal

∴ ∠PRQ = ∠PQR

Now, by Angle sum property, in △PRQ, ∠PRQ + ∠PQR + ∠RPQ = 1800

∵ ∠RPQ = 300

∴ ∠PRQ + ∠PRQ + 300 = 1800

∴ 2 ∠PRQ  + 300 = 1800

∴  ∠PRQ  = 750

Therefore, ∠PRQ= ∠PQR = 750

Since PQ ∥ SR, and RQ cuts these 2 lines:

∴ ∠PQR = ∠SRQ = 750   (Alternate angles)

Since PQ is tangent at Q and QR is chord at Q.

∴ ∠RSQ = ∠PQR = 75(∠RSQ in alternate segment of circle]

Now, In △SRQ,

∵ ∠RSQ + ∠SRQ + ∠SQR = 1800   (Angle sum property of a triangle)

∴ 75+ 75+ ∠SQR = 1800

∴  ∠SQR = 1800 – 150

∴  ∠SQR = 300

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The text passage contains a geometric problem involving tangents, chords, and angles in a circle, followed by a detailed step-by-step solution to find the measure of ∠RQS.
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Given a text passage, determine:
    1. If it contains a mathematical question (including word problems or calculations)
    2. If an answer to that mathematical question is provided in the text

First, describe the content structure you found in 1-2 sentences.

Then provide your analysis in the following format:
TYPE: <one of the following values>
- NO_QUESTION
- SINGLE_QUESTION
- SINGLE_QUESTION_WITH_ANSWER
- MULTIPLE_QUESTIONS
- MULTIPLE_QUESTIONS_WITH_ANSWERS
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Q) In the given figure, tangents PQ and PR are drawn to a circle such that ∠RPQ = 300. A chord RS is drawn parallel to tangent PQ. Find the ∠RQS.

Ans: In △PRQ, PQ and PR are tangents from an external point P to circle.

∴ PR = PQ

Since the angles opposites to equal sides are equal

∴ ∠PRQ = ∠PQR

Now, by Angle sum property, in △PRQ, ∠PRQ + ∠PQR + ∠RPQ = 1800

∵ ∠RPQ = 300

∴ ∠PRQ + ∠PRQ + 300 = 1800

∴ 2 ∠PRQ  + 300 = 1800

∴  ∠PRQ  = 750

Therefore, ∠PRQ= ∠PQR = 750

Since PQ ∥ SR, and RQ cuts these 2 lines:

∴ ∠PQR = ∠SRQ = 750   (Alternate angles)

Since PQ is tangent at Q and QR is chord at Q.

∴ ∠RSQ = ∠PQR = 75(∠RSQ in alternate segment of circle]

Now, In △SRQ,

∵ ∠RSQ + ∠SRQ + ∠SQR = 1800   (Angle sum property of a triangle)

∴ 75+ 75+ ∠SQR = 1800

∴  ∠SQR = 1800 – 150

∴  ∠SQR = 300

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user
The text passage contains a geometric problem involving tangents, chords, and angles in a circle, followed by a detailed step-by-step solution to find the measure of ∠RQS.

TYPE: SINGLE_QUESTION_WITH_ANSWER