Question

Triangle ABC is inscribed within a circle and has diameter line AB, with the inscribed angle at C. If the angle measure of A is 15 degree, what is the arc measure of AC?

__ degree

Answer 1

In a circle, when a triangle is inscribed such that one side is a diameter, the angle opposite that side is a right angle. This is a consequence of the Inscribed Angle Theorem.

For triangle \( ABC \), since \( AB \) is the diameter, angle \( C \) is a right angle (90 degrees). Given that angle \( A \) measures 15 degrees, we can find the measure of angle \( B \) using the triangle angle sum property, which states that the sum of the angles in a triangle is 180 degrees:

\[
\angle A + \angle B + \angle C = 180^\circ
\]
Substituting the known values:

\[
15^\circ + \angle B + 90^\circ = 180^\circ
\]
This simplifies to:

\[
\angle B + 105^\circ = 180^\circ
\]
Thus, we can solve for \( \angle B \):

\[
\angle B = 180^\circ - 105^\circ = 75^\circ
\]

Now, we want to find the arc measure of arc \( AC \). In a circle, the measure of an arc that subtends an inscribed angle is twice the angle measure. Thus, the measure of arc \( AC \) subtended by angle \( A \) will be:

\[
\text{Arc } AC = 2 \times \angle A = 2 \times 15^\circ = 30^\circ
\]

Therefore, the measure of arc \( AC \) is

\[
\boxed{30} \text{ degrees.}
\]