Question

g.gsr.8.1 (mc)

\\(\angle cab\\) is a central angle. if \\(m\widehat{cab} = 82^{\circ}\\), what is the measure of \\(\angle cab\\)?

\\(98^{\circ}\\)
\\(164^{\circ}\\)
\\(41^{\circ}\\)
\\(82^{\circ}\\)

Explanation:

Identify the given information

We are given a circle with center \(A\).
The angle \(\angle CAB\) is a central angle.
The measure of the intercepted minor arc \(\widehat{CB}\) (labeled as \(\widehat{CAB}\) in the text, representing the arc between \(C\) and \(B\) passing through the angle's opening) is \(82^\circ\).

Apply the Central Angle Theorem

By definition, the measure of a central angle in a circle is equal to the measure of its intercepted arc.
Therefore, we can set up the relationship:

$$ m\angle CAB = m\widehat{CB} $$

Calculate the angle measure

Using the given arc measure of \(82^\circ\):

$$ m\angle CAB = 82^\circ $$

Match with the given options

We compare our result with the multiple-choice options:

• \(98^\circ\)
• \(164^\circ\)
• \(41^\circ\)
• \(82^\circ\)

The correct option is \(82^\circ\).

Answer:

• (A) \(98^\circ\)
• (B) \(164^\circ\)
• (C) \(41^\circ\)
• (D) \(82^\circ\) (Correct answer)