Question

use the figure below to answer the following question.
triangle abc is inscribed in a circle o. what is the measure of ∠a?

Explanation:

Step1: Find the measure of arc \(ACB\)

The total measure of a circle is \(360^{\circ}\). Given arc \(AB = 100^{\circ}\), then arc \(ACB=360^{\circ}- 100^{\circ}=260^{\circ}\)

Step2: Use the property of inscribed angles subtended by a diameter

Since \(AC\) is a diameter, the arc \(AC\) is \(180^{\circ}\). Let's first recall the formula for the inscribed - angle theorem: the measure of an inscribed angle \(\theta\) is half the measure of its subtended arc \(s\), i.e., \(\theta=\frac{1}{2}s\)
The arc \(BC\) is \(260^{\circ}-180^{\circ} = 80^{\circ}\)

Step3: Calculate \(\angle A\)

\(\angle A\) is an inscribed angle subtended by arc \(BC\). By the inscribed - angle theorem \(\angle A=\frac{1}{2}\text{arc}(BC)\)
Since arc \(BC = 80^{\circ}\), then \(\angle A=\frac{1}{2}\times80^{\circ}=40^{\circ}\)

Answer:

\(40^{\circ}\)