Question

in diagram show below, \\( \overleftrightarrow { b d } \\) is tangent to the circle and \\( \angle b d g = 147 ^ { \circ } \\) what is \\( m \overparen { d g } \\)?

Explanation:

Step1: Find the measure of the angle between the tangent and the chord

We know that \(\angle BDG = 147^{\circ}\). The angle between the tangent \(\overleftrightarrow{BD}\) and the chord \(DG\) (let's call the angle \(\angle FDG\)) is \(180^{\circ}-\angle BDG\).

$$ \angle FDG=180^{\circ}- 147^{\circ}=33^{\circ} $$

Step2: Use the tangent - chord angle theorem

The measure of an angle formed by a tangent and a chord is half the measure of the intercepted arc. Let the measure of arc \(DG\) be \(m\overarc{DG}\). According to the tangent - chord angle theorem \(\angle FDG=\frac{1}{2}m\overarc{DG}\)

$$ m\overarc{DG} = 2\times\angle FDG $$

Substitute \(\angle FDG = 33^{\circ}\) into the formula:

$$ m\overarc{DG}=2\times33^{\circ}=66^{\circ} $$

Answer:

\(66^{\circ}\)