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

In a circle, when a triangle is inscribed such that one of its sides is the diameter, the angle opposite this side (i.e., at the point where the diameter meets the circle, which in this case is point C) is a right angle (90°).

Given angle A measures 15°, we can find the measure of angle C by using the fact that the sum of the angles in triangle ABC must be 180°:

\[
\text{Angle A} + \text{Angle B} + \text{Angle C} = 180°
\]

Let angle B be \( B \).

Thus, we have:

\[
15° + B + 90° = 180°
\]

This simplifies to:

\[
B + 105° = 180°
\]
\[
B = 180° - 105° = 75°
\]

Now, the angle measures in triangle ABC are:
- Angle A = 15°
- Angle B = 75°
- Angle C = 90°

The arc measure of arc AC corresponds to the angle subtended by that arc at the center of the circle. The inscribed angle theorem states that the measure of an inscribed angle is half the measure of the arc it intercepts.

Therefore, the arc AC is subtended by angle B at the center of the circle. The measure of arc AC is twice the measure of angle B:

\[
\text{Arc AC} = 2 \times \text{Angle B} = 2 \times 75° = 150°
\]

Thus, the arc measure of AC is \( \boxed{150°} \).