Question

point d is the in - center of triangle bca. if m∠fhg = 61°, what is the measure of ∠fdg?

Explanation:

Step1: Recall the property of angles in a circle

The measure of an inscribed - angle is half of the measure of the central - angle subtended by the same arc. Also, for a circle with center \(D\), \(\angle FHG\) and \(\angle FDG\) are related. The central - angle \(\angle FDG\) and the inscribed - angle \(\angle FHG\) subtend the same arc \(\overset{\frown}{FG}\).

Step2: Apply the inscribed - angle theorem

The inscribed - angle theorem states that if \(\angle FHG\) is an inscribed angle and \(\angle FDG\) is the central angle subtended by the same arc \(\overset{\frown}{FG}\), then \(m\angle FDG = 2m\angle FHG\).
Given \(m\angle FHG=61^{\circ}\), we substitute the value into the formula: \(m\angle FDG = 2\times61^{\circ}\).

Answer:

\(122^{\circ}\)