Angles In A Circle And Cyclic Quadrilateral
NIOS · Class 10 · Maths
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In a circle with center O, if an arc PQ subtends an angle of 80° at the center, what is the angle subtended by the same arc at a point R on the remaining part of the circle?
In a semicircle with diameter AB, if C is any point on the semicircle, what is the measure of angle ACB?
In a cyclic quadrilateral PQRS, if angle P = 75°, what is the measure of angle R?
If two chords AB and CD of a circle are equal, and angle AOB = 60° (where O is the center), what is angle COD?
Sample Questions
In circle with center O, if arc PQ has measure 120°, and points R and S are on the remaining part of circle, what can you say about angles PRQ and PSQ?
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They are equal
Step 1: Arc PQ subtends angles at two different points R and S on the circle. Step 2: Both R and S are on the same segment (remaining part of circle). Step 3: By the theorem 'angles in the same segment are equal', angle PRQ = angle PSQ. Step 4: Both angles equal 120°/2 = 60°. This is regardless of where R and S are positioned on the same segment.
If angle subtended by an arc at center is 140°, what is the angle subtended by the same arc at any point on the circle?
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70°
Step 1: Given central angle = 140°. Step 2: Apply the inscribed angle theorem. Step 3: Inscribed angle = (1/2) × central angle = (1/2) × 140° = 70°. Step 4: This relationship always holds for any arc and any point on the remaining part of the circle.
In a circle, chord AB subtends angle 45° at point C on the circle. What angle does chord AB subtend at the center O?
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90°
Step 1: Chord AB subtends 45° at point C (inscribed angle). Step 2: By central angle theorem, central angle = 2 × inscribed angle. Step 3: Central angle AOB = 2 × 45° = 90°. Step 4: Therefore, chord AB subtends 90° at center O.
A quadrilateral PQRS has angle P = 95° and angle R = 85°. Can this quadrilateral be cyclic?
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Yes, because P + R = 180°
Step 1: For a quadrilateral to be cyclic, opposite angles must be supplementary. Step 2: P and R are opposite angles. Check: P + R = 95° + 85° = 180°. Step 3: Since opposite angles sum to 180°, the quadrilateral can be cyclic. Step 4: We need to verify the other pair of opposite angles (Q and S) also sum to 180° for confirmation.
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