Question

circle g is shown in the diagram below.
complete the paragraph.
in circle g, ∠dgf is and ∠def is
measure of ∠def is the measure of ∠dgf.

Explanation:

To solve this, we use circle theorems:

Step 1: Identify ∠DGF

∠DGF has its vertex at the center \( G \) of the circle, with sides \( GD \) and \( GF \) (both radii). By definition, a central angle has its vertex at the circle’s center. Thus, \( \angle DGF \) is a central angle.

Step 2: Identify ∠DEF

∠DEF has its vertex at point \( E \) on the circle, with sides \( ED \) and \( EF \) (chords of the circle). By definition, an inscribed angle has its vertex on the circle and its sides as chords. Thus, \( \angle DEF \) is an inscribed angle.

Step 3: Relate their measures

The central angle theorem states that an inscribed angle is half the measure of its intercepted central angle. For \( \angle DEF \) (inscribed) and \( \angle DGF \) (central, intercepting the same arc \( \overset{\frown}{DF} \)), the measure of \( \angle DEF \) is \( \frac{1}{2} \) (or half) the measure of \( \angle DGF \).

Answer:

In circle \( G \), \( \angle DGF \) is a \(\boldsymbol{\text{central angle}}\) and \( \angle DEF \) is an \(\boldsymbol{\text{inscribed angle}}\). The measure of \( \angle DEF \) is \(\boldsymbol{\text{half}}\) the measure of \( \angle DGF \).