Question

A circle has a diameter AB and an inscribed angle at vertex C forming triangle ABC. The measure of angle A is 27°. Using what you know about inscribed angles, what is the angle measure of B?(1 point)m∠B=º

Answer 1

In a circle, the inscribed angle is half of the measure of the arc that it intercepts. Since angle C is inscribed and intercepts the arc AB, we can use the relationship between the inscribed angles and the angles at the circumference.

Since AB is the diameter, arc AB corresponds to the semicircle and thus measures 180°.

Given that angle A measures 27°, we know that angle C, which intercepts arc AB, is equal to half of the measure of arc AB. The relationship of angles in triangle ABC is given by:

\[
m\angle A + m\angle B + m\angle C = 180°
\]

Let \(m\angle B\) be \(x\). Thus, we can express this as:

\[
27° + x + m\angle C = 180°
\]

Now, since angle C is inscribed and intercepts arc AB, we know that:

\[
m\angle C = \frac{1}{2} \cdot m \text{(arc AB)} = \frac{1}{2} \cdot 180° = 90°
\]

Now we substitute \(m \angle C\) into our triangle angle sum equation:

\[
27° + x + 90° = 180°
\]

Combining the known angle measures gives us:

\[
117° + x = 180°
\]

To find \(x\), we subtract 117° from both sides:

\[
x = 180° - 117° = 63°
\]

Thus, the measure of angle B is:

\[
m\angle B = 63°
\]

So, \(m\angle B = 63°\).