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 (square)
degrees. since the measure of a straight angle is (180^{circ})
the measure of angle (square) must also be (90^{circ}) by the
(square). a bisector cuts the angle
(5^{circ})
angle addition postulate
reflexive property
segment addition property
symmetric property

Explanation:

Step1: Determine the measure of \(\angle CED\)

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

Step2: Use the angle addition postulate

We know that \(\angle AED\) is a straight angle (\(\angle AED=180^{\circ}\)). By the angle addition postulate \(\angle AED=\angle AEC+\angle CED\). Since \(\angle CED = 90^{\circ}\), then \(\angle AEC=\angle AED-\angle CED\). Substituting the values, \(\angle AEC=180^{\circ}- 90^{\circ}=90^{\circ}\)

Step3: Use the definition of an angle bisector

Since \(\overrightarrow{EB}\) bisects \(\angle AEC\), by the definition of an angle bisector \(\angle AEB=\frac{1}{2}\angle AEC\). Substituting \(\angle AEC = 90^{\circ}\), we get \(\angle AEB=\frac{1}{2}\times90^{\circ}=45^{\circ}\)

Answer:

First blank: \(90\); Second blank: \(AEC\); Third blank: \(angle\ addition\ postulate\)