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°, what is the arc measure of AC?(1 point)º

Answer 1

Since triangle ABC is inscribed in the circle, and AB is the diameter of the circle, angle C is a right angle (90°) due to the inscribed angle theorem, which states that an angle inscribed in a semicircle is a right angle.

Given that angle A is 15°, we can find angle B in triangle ABC:

\[
\text{Angle C} = 90° \\
\text{Angle A} + \text{Angle B} + \text{Angle C} = 180° \\
15° + \text{Angle B} + 90° = 180° \\
\text{Angle B} = 180° - 15° - 90° = 75°
\]

Now, to find the arc measure of AC, we can use the relationship between the inscribed angle and the arc it subtends. The inscribed angle is half of the measure of the arc that it subtends:

\[
\text{Arc AC} = 2 \cdot \text{Angle B}
\]

Substituting the value of angle B:

\[
\text{Arc AC} = 2 \cdot 75° = 150°
\]

Thus, the arc measure of AC is:

\[
\boxed{150^\circ}
\]