Question

ac is tangent to circle o at point a, and m\widehat{ab}=58. what is m\angle acb?

Explanation:

Step1: Use the property of tangent to circle

Since \(AC\) is tangent to circle \(O\) at \(A\), then \(OA\perp AC\), so \(\angle OAC = 90^{\circ}\).

Step2: Find the measure of \(\angle AOB\)

Given \(m\overarc{AB}=58^{\circ}\), and the central - angle theorem states that the measure of a central angle is equal to the measure of its intercepted arc. So \(m\angle AOB = 58^{\circ}\).

Step3: Use the angle - sum property of a triangle

In \(\triangle AOC\), we know that the sum of interior angles of a triangle is \(180^{\circ}\). Let \(m\angle ACB=x\).
We have \(\angle OAC = 90^{\circ}\), \(\angle AOB\) is an exterior angle of \(\triangle AOC\) (by the exterior - angle theorem of a triangle: \(\angle AOB=\angle OAC+\angle ACB\)).
Substituting the values: \(58^{\circ}=90^{\circ}-x\) (because \(\angle AOB\) and \(\angle AOC\) are related, and using the fact that in right - triangle \(OAC\) with \(\angle OAC = 90^{\circ}\), we can also use the formula for the angle between a tangent and a secant: \(m\angle ACB=\frac{1}{2}(m\overarc{AB})\) (the measure of an angle formed by a tangent and a secant is half the measure of the intercepted arc).
\(m\angle ACB=\frac{1}{2}\times58^{\circ}=29^{\circ}\)

Answer:

\(29^{\circ}\) (Option D)