Question

what is the measure of ∠cae? 135° 55° a. 25° b. 55° c. 80° d. 40°

Explanation:

Step1: Recall the inscribed - angle theorem

The measure of an inscribed angle is half the measure of its intercepted arc. The measure of the inscribed angle $\angle BDE = 55^{\circ}$, and the measure of arc $CE=135^{\circ}$.

Step2: Use the formula for the measure of an inscribed - angle related to the difference of arcs

The measure of $\angle CAE$ is given by $\frac{1}{2}(\text{measure of arc }CE-\text{measure of arc }BD)$. First, we know that the measure of the inscribed angle $\angle BDE$ intercepts arc $BE$. By the inscribed - angle theorem, if $\angle BDE = 55^{\circ}$, then the measure of arc $BE = 2\times\angle BDE=110^{\circ}$.
Let the measure of arc $BD=x$. Then, since arc $BE$ is composed of arc $BD$ and arc $DE$, and we know arc $BE = 110^{\circ}$. Also, the whole - circle is $360^{\circ}$.
We know that the measure of $\angle CAE=\frac{1}{2}(\text{measure of arc }CE - \text{measure of arc }BD)$.
The measure of arc $CE = 135^{\circ}$. Since the measure of the inscribed angle $\angle BDE = 55^{\circ}$, the measure of arc $BE=110^{\circ}$. Assume arc $BD$ has measure $x$.
We know that the measure of $\angle CAE=\frac{1}{2}(135 - 55)$.

Step3: Calculate the measure of $\angle CAE$

$\angle CAE=\frac{1}{2}(135 - 55)=\frac{1}{2}\times80 = 40^{\circ}$

Answer:

D. $40^{\circ}$