Question

in the diagram shown below, \\( \overleftrightarrow { b d } \\) is tangent to the circle and \\( m \overparen { d g } = 64 ^ { \circ } \\).

what is the measure of \\( \angle b d g \\)?

\\( 99 ^ { \circ } \\)
\\( 148 ^ { \circ } \\)
\\( 136 ^ { \circ } \\)
\\( 128 ^ { \circ } \\)

Explanation:

Step1: Recall the tangent - secant angle formula

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. The formula is \(m\angle BDG=\frac{1}{2}(m\overarc{DG})\) (this is incorrect, the correct formula is \(m\angle BDG = \frac{1}{2}(360 - m\overarc{DG})\) when the angle is formed by a tangent and a chord and we consider the major arc). The measure of the major arc \(m\overarc{DG}\) (the arc that is not the \(64^{\circ}\) arc) is \(360 - 64=296^{\circ}\).

Step2: Calculate the measure of \(\angle BDG\)

Using the formula \(m\angle BDG=\frac{1}{2}(360 - m\overarc{DG})\), substitute \(m\overarc{DG} = 64^{\circ}\). Then \(m\angle BDG=\frac{1}{2}(360 - 64)=\frac{1}{2}\times296 = 148^{\circ}\)

Answer:

\(148^{\circ}\)