Question

lines cd and de are tangent to circle a. if ce is 112°, what is the measure of ∠cde?

Explanation:

Step1: Recall the property of tangents and central - arc

The measure of the angle formed by two tangents to a circle is half the difference of the measures of the intercepted arcs. The sum of the measures of the major and minor arcs of a circle is 360°. Given the measure of the major arc $\widehat{CE}=112^{\circ}$, then the measure of the minor arc $\widehat{CB E}=360 - 112=248^{\circ}$.

Step2: Apply the tangent - angle formula

The formula for the measure of the angle $\angle CDE$ formed by two tangents $CD$ and $DE$ is $\angle CDE=\frac{1}{2}(\text{major arc}-\text{minor arc})$. Substitute the values of the major arc ($248^{\circ}$) and the minor arc ($112^{\circ}$) into the formula: $\angle CDE=\frac{1}{2}(248 - 112)$.

Step3: Calculate the value of $\angle CDE$

First, calculate the difference inside the parentheses: $248-112 = 136$. Then, divide by 2: $\frac{136}{2}=68^{\circ}$.

Answer:

$68^{\circ}$