Question

angle g is a circumscribed angle of circle e. what is the measure of angle g, in terms of x? x° + x° x° + 90° 180° - x° 180° - 2x°

Explanation:

Step1: Use the property of tangent - radius

Since \(GF\) and \(GD\) are tangents to the circle \(E\), \(EF\perp GF\) and \(ED\perp GD\), so \(\angle EFG = 90^{\circ}\) and \(\angle EDG=90^{\circ}\).

Step2: Use the sum of interior angles of a quadrilateral

The sum of interior angles of quadrilateral \(EFGD\) is \(360^{\circ}\). Let \(\angle G=y\). Then \(\angle EFG+\angle FGD+\angle GDE+\angle DEF = 360^{\circ}\). Substituting \(\angle EFG = 90^{\circ}\), \(\angle EDG = 90^{\circ}\), \(\angle DEF=2x^{\circ}\) (given \(\angle FED=x^{\circ}+x^{\circ}\)) and \(\angle FGD = y\) into the formula: \(90 + y+90 + 2x=360\).

Step3: Solve for \(y\)

Simplify the equation \(90 + y+90 + 2x=360\) to \(y+180 + 2x=360\). Then \(y=360-(180 + 2x)=180 - 2x\).

Answer:

\(180^{\circ}-2x^{\circ}\)