Question

21
select the correct answer from each drop - down menu.
the center of the circle is at o. segment ab is tangent to circle o at point d.
the measure of ∠ado is 90 °.
the measure
only angle dge
only angle cgf is
both angles cgd and egf are
both angles cgf and dge are
equal to \\(\frac{1}{2}(\text{m}\overarc{cf} + \text{m}\overarc{de})\\).

Explanation:

Step1: Recall tangent-radius property

A tangent to a circle is perpendicular to the radius at the point of tangency. So $\angle ADO = 90^\circ$, which matches the given value.

Step2: Identify angle-chord theorems

For angles formed by two intersecting chords: the measure of the angle is half the sum of the measures of the intercepted arcs.

• $\angle DGE$ is formed by intersecting chords $DF$ and $CE$. It intercepts arcs $\widehat{DE}$ and $\widehat{CF}$, so $m\angle DGE = \frac{1}{2}(m\widehat{CF} + m\widehat{DE})$.
• $\angle CGF$ is vertical to $\angle DGE$, so it has the same measure: $m\angle CGF = \frac{1}{2}(m\widehat{CF} + m\widehat{DE})$.
• $\angle CGD$ and $\angle EGF$ intercept arcs $\widehat{CD}$ and $\widehat{EF}$, so their measure is $\frac{1}{2}(m\widehat{CD} + m\widehat{EF})$, which does not match the given expression.

Answer:

The measure of $\angle ADO$ is $90^\circ$.
both angles CGF and DGE are equal to $\frac{1}{2}(m\widehat{CF} + m\widehat{DE})$.