Question

Lines CD and CE are tangent to circle A.If m∠DAE = 142°, what is the measure of ∠DCE?

Answer 1

Given that lines CD and CE are tangent to circle A, you know that the measure of the Central Angle is:

$m\angle DAE=142°$

By definition, the measure of a Central Angle is equal to the measure of the corresponding Intercepted Arc. Therefore, you can determine that:

$m\angle DAE=mDE=142°$

By definition, the angle formed by the intersection of two tangents outside a circle can be calculated by subtracting the measure of the Intercepted Minor Arc from the measure of the Intercepted Major Arc and dividing the Difference by 2.

Knowing that there are 360 degrees in a circle, you can determine that the measure of the Major Intercepted Arc is:

$Major=360\text{\degree}-142\text{\degree}=218\text{\degree}$

Therefore, you can determine that:

$m\angle DCE=\frac{218\text{\degree}-142\text{\degree}}{2}=38\text{\degree}$

Hence, the answer is: Second option.