6. In given fig., PQ is the diameter of the circle. If PQR =65, RPS = 25° and QPT = 60°, then
find the measure of (i) QPR () PRS () PSR (iv) PQT
Answers
The measure of
(i)QPR = 25°
(ii)PRS = 40°
(iii)PSR = 115°
(iv)PQT = 30°
Step-by-step explanation:
Referring to the figure attached below, we will solve for the measure of ∠QPR, ∠PRS, ∠PSR & ∠PQT:
Step 1: Finding the angle QPR
PQ is given the diameter of the circle
∴ ∠PRQ = ∠PTQ = 90° ….. [angle in a semicircle is 90°] …. (i)
In ∆ PRQ, applying the angle sum property,
∠PRQ + ∠PQR + ∠QPR = 180°
⇒ ∠QPR + 90° + 65° = 180° …… [∠PRQ = 90° (from (i)) & ∠PQR = 65° (given)]
⇒ ∠QPR = 180° - (90° + 65°)
⇒ ∠QPR = 25°
Step 2: Finding the angle PSR
Considering the cyclic quadrilateral PSRQ, we have
∠RQP + ∠PSR = 180° ….. [sum of opposite angles of a cyclic quadrilateral is 180°]
⇒ ∠PSR = 180° - 65° …… [value of angle PSR = 65° (given)]
⇒ ∠PSR = 115°
Step 3: Find angle PRS
In ∆ PSR, applying the angle sum property,
∠PSR + ∠PRS + ∠RPS = 180°
⇒ ∠PRS = 180° - (115° + 25°)
⇒ ∠PRS = 40°
Step 4: Finding angle PQT
In ∆ PTQ, applying the angle sum property,
∠QPT + ∠PQT + ∠PTQ = 180°
⇒ 60° + 90° + angle PTQ = 180° …….. [value of ∠QPT = 60° (given) and ∠PQT = 90°]
⇒ ∠PTQ = 180° - (60° + 90°)
⇒ ∠PTQ = 30°
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Answer:
The answer is attached as images
Step-by-step explanation:
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