Question

for the circle with center a: if ( mwidehat{db}=72^{circ} ) what are the measures of ( angle dab ) and ( angle dfb )? 6 of 6 question ( mangle dab = 72^{circ} ) and ( mangle dfb = 144^{circ} ) ( mangle dab = 144^{circ} ) and ( mangle dfb = 72^{circ} ) ( mangle dab = 72^{circ} ) and ( mangle dfb = 36^{circ} ) ( mangle dab = 36^{circ} ) and ( mangle dfb = 72^{circ} )

Explanation:

Step1: Recall the central - angle and inscribed - angle relationship

The measure of a central angle (\(\angle DAB\)) is equal to the measure of its intercepted arc (\(\overset{\frown}{DB}\)). The measure of an inscribed angle (\(\angle DFB\)) is half of the measure of its intercepted arc (\(\overset{\frown}{DB}\)).

Step2: Calculate \(\angle DAB\)

Since \(\angle DAB\) is a central angle and \(m\overset{\frown}{DB}=72^{\circ}\), then \(m\angle DAB = 72^{\circ}\) (central angle - arc relationship: \(m\angle\theta=m\overset{\frown}{s}\), where \(\theta\) is the central angle and \(s\) is the intercepted arc).

Step3: Calculate \(\angle DFB\)

Since \(\angle DFB\) is an inscribed angle and \(m\overset{\frown}{DB}=72^{\circ}\), using the formula \(m\angle\alpha=\frac{1}{2}m\overset{\frown}{s}\) (where \(\alpha\) is the inscribed angle and \(s\) is the intercepted arc), we have \(m\angle DFB=\frac{1}{2}\times72^{\circ}=36^{\circ}\)

Answer:

\(m\angle DAB = 72^{\circ}\) and \(m\angle DFB = 36^{\circ}\), so the third option \(m\angle DAB = 72^{\circ}\) and \(m\angle DFB = 36^{\circ}\) is correct.