Question

if \\( \overarc{ce} \\) is \\( 125^{\circ} \\), what is the measure of \\( \angle cde \\)? \\( 55^{\circ} \\) \\( 62.5^{\circ} \\) \\( 117.5^{\circ} \\) \\( 125^{\circ} \\)

Explanation:

Step1: Recall the formula for the measure of an angle formed by two tangents

The measure of an angle formed by two tangents drawn to a circle is half the difference of the measures of the intercepted arcs. The formula is \( \angle CDE=\frac{1}{2}(360^{\circ}-\text{measure of arc } CE - \text{measure of arc } CE)\). Wait, no, more precisely, the measure of an angle formed outside the circle by two tangents is \( \angle CDE=\frac{1}{2}(m\overset{\frown}{major\ arc\ CE}-m\overset{\frown}{minor\ arc\ CE})\). Since the sum of the measures of a major arc and a minor arc of a circle is \(360^{\circ}\), and \(m\overset{\frown}{minor\ arc\ CE} = 125^{\circ}\), then \(m\overset{\frown}{major\ arc\ CE}=360^{\circ}- 125^{\circ}=235^{\circ}\).

Step2: Apply the formula

Substitute into the formula \( \angle CDE=\frac{1}{2}(m\overset{\frown}{major\ arc\ CE}-m\overset{\frown}{minor\ arc\ CE})\). So \( \angle CDE=\frac{1}{2}(235^{\circ}-125^{\circ})\).

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Answer:

\(55^{\circ}\)