In the given figure PAQ is the tangent of the circle at point A and ABCD is a…

When a tangent touches a circle at a point and a chord is drawn from that same point, the tangent-chord angle equals the inscribed angle that the same chord subtends in the alternate segment. Also, in any cyclic quadrilateral, each pair of opposite angles adds up to 180°.

1. The tangent ray AQ and chord AC meet at A on the side of the circle containing arc ABC, so ∠CAQ = 70° is the tangent-chord angle for chord AC on that side.

2. By the tangent-chord angle theorem, this angle equals the inscribed angle that chord AC subtends from the opposite arc, i.e. ∠ADC = 70°.

3. ABCD is a cyclic quadrilateral, so ∠ABC and ∠ADC are opposite angles and satisfy ∠ABC + ∠ADC = 180°.

4. Substituting ∠ADC = 70°: ∠ABC = 180° − 70° = 110°.

As an independent cross-check using arcs:

1. Writing the four arcs AB, BC, CD, DA as a, b, c, d (a + b + c + d = 360°): the tangent-chord angle ∠CAQ = (a+b)/2 = 70°, so a + b = 140°.

2. ∠ABC is the inscribed angle subtending arc ADC = c + d = 360° − 140° = 220°, so ∠ABC = 220°/2 = 110°, confirming the result.

Hence ∠ABC = 110°.