Question

consider the diagram of the circle shown below. the lines \\( \overleftrightarrow { d g } \\) and \\( \overleftrightarrow { g b } \\) are tangent to the circle and intersect at point g. which equation shows the relationship between the measures of \\( \angle b g d \\) and one of the arcs intersected by the tangent lines? 4 of 6 question \\( 2 ( m \angle b g d ) = 360 ^ { \circ } - 2 ( m \overparen { b c d } ) \\) \\( m \angle b g d = 360 ^ { \circ } - m \overparen { b d } \\) \\( m \angle b g d = 360 ^ { \circ } - m \overparen { b c d } \\) \\( 2 ( m \angle b g d ) = 360 ^ { \circ } - 2 ( m \overparen { b d } ) \\)

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 from an external point to a circle is half the difference of the measures of the intercepted arcs. The formula is \(m\angle BGD=\frac{1}{2}(m\overset{\frown}{BCD}-m\overset{\frown}{BD})\). Also, we know that \(m\overset{\frown}{BCD}+m\overset{\frown}{BD} = 360^{\circ}\), so \(m\overset{\frown}{BCD}=360^{\circ}-m\overset{\frown}{BD}\).
Substitute \(m\overset{\frown}{BCD}\) into the angle - tangent formula:

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Answer:

\(2(m\angle BGD)=360^{\circ}-2(m\overset{\frown}{BD})\) (the fourth option)