Question

angle a is circumscribed about circle o,
what is the measure of ∠a?
□°
(there is a diagram with circle o, points c, b on the circle, and quadrilateral oca b with angle at o being 92°)

Explanation:

Step1: Recall tangent properties

Tangents to a circle from a common external point are perpendicular to the radius at the point of contact. So, \( \angle OCA = 90^\circ \) and \( \angle OBA = 90^\circ \).

Step2: Use quadrilateral angle sum

In quadrilateral \( OCA B \), the sum of interior angles is \( 360^\circ \). Let \( \angle A = x \), \( \angle OCA = 90^\circ \), \( \angle OBA = 90^\circ \), and \( \angle COB = 92^\circ \). Then:

$$ x + 90^\circ + 92^\circ + 90^\circ = 360^\circ $$

Step3: Solve for \( x \)

Simplify the equation:

$$ x + 272^\circ = 360^\circ $$

Subtract \( 272^\circ \) from both sides:

$$ x = 360^\circ - 272^\circ = 88^\circ $$

Answer:

\( 88^\circ \)