Question

identifying relationships from diagrams
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 degrees. since the measure of a straight angle is (180^{circ}), the measure of angle must also be (90^{circ}) by the. a bisector cuts the angle measure in half. (mangle aeb) is (45^{circ}).

Explanation:

Step1: Determine the measure of ∠CED

A right - angle measures \(90^{\circ}\). So, \(\angle CED = 90^{\circ}\).

Step2: Use the property of a straight - angle

Since \(\angle AED\) is a straight - angle (\(\angle AED=180^{\circ}\)), and \(\angle AED=\angle AEC+\angle CED\). Substituting \(\angle CED = 90^{\circ}\), we get \(\angle AEC+\ 90^{\circ}=180^{\circ}\). By the subtraction property of equality, \(\angle AEC=180^{\circ}-90^{\circ} = 90^{\circ}\).

Step3: Use the angle - bisector property

Since \(\overrightarrow{EB}\) bisects \(\angle AEC\), by the definition of an angle bisector (if a ray bisects an angle, it divides the angle into two equal parts), \(\angle AEB=\frac{1}{2}\angle AEC\). Substituting \(\angle AEC = 90^{\circ}\), we have \(\angle AEB=\frac{1}{2}\times90^{\circ}=45^{\circ}\).

Answer:

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 \(\angle AEC\) must also be \(90^{\circ}\) by the angle - addition postulate. A bisector cuts the angle measure in half. \(m\angle AEB\) is \(45^{\circ}\).