Question

in the given figure AB is a chord of circle, and PQ is a tangent at point B of the circle. if angle AOB = 110° , THEN FIND angle ABQ.​

Answer 1

$\large\underline{\sf{Given- }}$

• A circle with centre O

such that

• AB is chord of circle.

• PQ is tangent to a circle at B.

• ∠ AOB = 110°

$\large\underline{\sf{To\:Find - }}$

• measure of ∠ABQ

$\large\underline{\sf{Solution-}}$

We know that,

• Angle subtended at the centre of a circle by an arc is double the angle subtended at circumference.

• ⇛ ∠ AOB = 2 ∠ACB

• ⇛ 110° = 2 ∠ ACB

• ⇛ ∠ ACB = 55°.

Now,

• PQ is a tangent to a circle at B.

We know,

• Angle in Alternate Segment :- The angle between a chord of a circle and a tangent through any of the chord end-point is equal to the angle in the alternate segment.

• ⇛∠ABQ = ∠ACB

• ⇛ ∠ABQ = 55°

$\overbrace{ \underline { \boxed { \bf \therefore \: \angle \: ABQ \: = \: 55 \degree}}}$

Additional Information :-

• 1. Angle in same segments are equal.

• 2. Angle in semicircle is 90°.

• 3. Sum of pair of opposite angles of a cyclic quadrilateral is supplementary.

• 4. The exterior angle of a cyclic quadrilateral is equals to interior opposite angle.

• 5. Perpendicular drawn from centre bisects the chord.

• 6. Length of tangents drawn from external point are equal.