Question

in circle e with ( mangle dgf = 41^{circ} ), find the ( mangle def ).

Explanation:

Step1: Recall the central - inscribed angle relationship

The measure of an inscribed angle is half the measure of the central angle that subtends the same arc.
In circle \(E\), \(\angle DGF\) is an inscribed angle and \(\angle DEF\) is a central angle. They both subtend arc \(DF\).
The formula is \(m\angle DEF = 2\times m\angle DGF\).

Step2: Substitute the given value

Given \(m\angle DGF=41^{\circ}\).
Substitute into the formula: \(m\angle DEF = 2\times41^{\circ}\).

$$m\angle DEF=82^{\circ}$$

Answer:

\(82^{\circ}\)