In the diagram, angle EDF = 58 degrees. Find the measure of arc ERF.

Solution:

• In the diagram, ED and FD are tangents to the circle at points E and F respectively. D is the external point.
• The angle formed by two tangents drawn from an external point to a circle is equal to half the difference of the measures of the intercepted arcs. The intercepted arcs are the major arc ERF and the minor arc EF.
• However, the angle $$\angle EDF$$ intercepts the arc EF. The formula for the angle formed by two tangents from an external point is: $$\angle EDF = \frac{1}{2} (\widehat{major ext{ }EF} - \widehat{minor ext{ }EF})$$.
• Let $$\widehat{EF}$$ be the measure of the minor arc EF. Then the measure of the major arc ERF is $$360^{\circ} - \widehat{EF}$$.
• So, the formula is $$\angle EDF = \frac{1}{2} ((360^{\circ} - \widehat{EF}) - \widehat{EF})$$.
• $$\angle EDF = \frac{1}{2} (360^{\circ} - 2 \widehat{EF})$$.
• $$\angle EDF = 180^{\circ} - \widehat{EF}$$.
• We are given $$\angle EDF = 58^{\circ}$$.
• So, $$58^{\circ} = 180^{\circ} - \widehat{EF}$$.
• $$\widehat{EF} = 180^{\circ} - 58^{\circ} = 122^{\circ}$$.
• The question asks for the measure of arc ERF. This refers to the major arc.
• The major arc ERF is $$360^{\circ} - \widehat{EF}$$.
• Major arc ERF = $$360^{\circ} - 122^{\circ} = 238^{\circ}$$.
• Let's double check the formula for the angle between two tangents. Yes, the 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 intercepted arcs are the major arc and the minor arc.
• So, $$\angle EDF = 58^{\circ}$$. Let the minor arc EF be $$x$$. Then the major arc ERF is $$360 - x$$.
• The formula is $$\angle EDF = \frac{1}{2} ( ext{far arc} - ext{near arc})$$. Here, the far arc is the major arc ERF, and the near arc is the minor arc EF.
• So, $$58^{\circ} = \frac{1}{2} ((360^{\circ} - \widehat{EF}) - \widehat{EF})$$.
• $$58^{\circ} = \frac{1}{2} (360^{\circ} - 2 imes \widehat{EF})$$.
• $$58^{\circ} = 180^{\circ} - \widehat{EF}$$.
• $$\widehat{EF} = 180^{\circ} - 58^{\circ} = 122^{\circ}$$.
• The question asks for the measure of arc ERF, which is the major arc.
• Major arc ERF = $$360^{\circ} - ext{minor arc } EF = 360^{\circ} - 122^{\circ} = 238^{\circ}$$.
• The red 'X' on the arc ERF might be highlighting this arc.

Final Answer:

The final answer is $$\boxed{238^{\circ}}$$