angle bcd is a circumscribed angle of circle a. angle bac measures 53°. what is the measure of angle bcd…

Explanation:

Step1: Recall the property of circum - scribed angle

The measure of a circum - scribed angle and the central angle that subtends the same arc are supplementary.

Step2: Identify the central angle and circum - scribed angle

The central angle $\angle BAC = 53^{\circ}$, and $\angle BCD$ is the circum - scribed angle.

Step3: Calculate the measure of $\angle BCD$

Let $m\angle BCD=x$. We know that $x + m\angle BAC=180^{\circ}$. Substituting $m\angle BAC = 53^{\circ}$, we get $x=180^{\circ}-53^{\circ}=127^{\circ}$. But there is a wrong in the above. In fact, if we consider the relationship between the central angle and the circum - scribed angle formed by two tangents to a circle from an external point. The measure of the circum - scribed angle is supplementary to the central angle that subtends the arc intercepted by the tangents. The correct relationship is that the measure of the circum - scribed angle $\angle BCD$ and the central angle $\angle BAD$ (where $\angle BAD$ is the central angle for the arc intercepted by the tangents $BC$ and $CD$) are supplementary. Since $\angle BAC = 53^{\circ}$, the central angle for the arc intercepted by the tangents is $2\times53^{\circ}=106^{\circ}$. Let the measure of $\angle BCD$ be $y$. Then $y + 106^{\circ}=180^{\circ}$. $y=180^{\circ}-106^{\circ}=74^{\circ}$

Answer:

$74^{\circ}$