Lines CD and DE are tangent to circle A:

Question

Lines CD and DE are tangent to circle A and intersect at point D. Arc CE measures 112 degrees. Point B lies on circle A.

If arc CE is 112°, what is the measure of ∠CDE?

124°
136°
68°
56°

Answer 1

To find the measure of angle \( \angle CDE \), we can use the property of tangents and the arcs they subtend in a circle.

Given that lines \( CD \) and \( DE \) are tangent to circle \( A \) at points \( C \) and \( E \) respectively, and that arc \( CE \) measures \( 112^\circ \), we can find \( \angle CDE \) using the following relationship:

The angle formed by two tangent lines (in this case, \( CD \) and \( DE \)) that intersect outside the circle equals half the measure of the intercepted arc. Here, the intercepted arc is \( CE \).

Therefore, we can calculate:

\[ \angle CDE = \frac{1}{2} \times \text{measure of arc } CE \] \[ \angle CDE = \frac{1}{2} \times 112^\circ = 56^\circ \]

Thus, the measure of \( \angle CDE \) is \( 56^\circ \).

So the answer is \( \boxed{56^\circ} \).