Question

given: $overrightarrow{eb}$ bisects $angle aec$. $angle aed$ is a straight angle. prove: $mangle aeb = 45^{circ}$ complete the paragraph proof. we are given that $overrightarrow{eb}$ bisects $angle aec$. from the diagram, $angle ced$ is a right angle, which measures 90 degrees. since the measure of a straight angle is 180^{circ}, the measure of angle must also be 90 by the a bisector cuts the angle measure in half. $mangle aeb$ is 45^{circ}.

Explanation:

Step1: Identify straight - angle property

Since $\angle AED$ is a straight angle, $m\angle AED = 180^{\circ}$. And $\angle CED = 90^{\circ}$. By the angle - addition postulate, $m\angle AEC+m\angle CED=m\angle AED$. So $m\angle AEC=180^{\circ}- 90^{\circ}=90^{\circ}$.

Step2: Use angle - bisector property

Given that $\overrightarrow{EB}$ bisects $\angle AEC$. By the definition of an angle bisector, it divides an angle into two equal parts. So $m\angle AEB=\frac{1}{2}m\angle AEC$. Since $m\angle AEC = 90^{\circ}$, then $m\angle AEB=\frac{1}{2}\times90^{\circ}=45^{\circ}$.

Answer:

The measure of $\angle AEB$ is $45^{\circ}$ because $\angle AED = 180^{\circ}$, $\angle CED = 90^{\circ}$, so $\angle AEC=90^{\circ}$, and since $\overrightarrow{EB}$ bisects $\angle AEC$, $m\angle AEB = 45^{\circ}$.