Question

given: (mangle aeb = 45^{circ}), (angle aec) is a right - angle. prove: (overrightarrow{eb}) bisects (angle aec). proof: we are given that (mangle aeb = 45^{circ}) and (angle aec) is a right angle. the measure of (angle aec) is (90^{circ}) by the definition of a right angle. applying the segment addition property gives (mangle aeb + mangle bec=90^{circ}). the substitution gives (45^{circ}+mangle bec = 90^{circ}). the subtraction gives (mangle bec = 45^{circ}), so (angle aeb) and (angle bec) have the same measure. (overrightarrow{eb}) divides (angle aec) into two congruent angles, making it an angle bisector. segment addition property, angle addition postulate, reflexive property, transitive property

Explanation:

Step1: Recall right - angle definition

Since $\angle AEC$ is a right angle, $m\angle AEC = 90^{\circ}$ by the definition of a right angle.

Step2: Apply angle - addition postulate

We know that $\angle AEC=\angle AEB+\angle BEC$ by the angle addition postulate.

Step3: Substitute known values

Given $m\angle AEB = 45^{\circ}$ and $m\angle AEC = 90^{\circ}$, substituting into $\angle AEC=\angle AEB+\angle BEC$ gives $90^{\circ}=45^{\circ}+m\angle BEC$.

Step4: Solve for $m\angle BEC$

Using the subtraction property of equality, we subtract $45^{\circ}$ from both sides: $m\angle BEC=90^{\circ}- 45^{\circ}=45^{\circ}$.

Step5: Determine angle - bisector

Since $m\angle AEB = 45^{\circ}$ and $m\angle BEC = 45^{\circ}$, $\overrightarrow{EB}$ bisects $\angle AEC$ (a ray that divides an angle into two congruent angles is an angle - bisector).

Answer:

The missing term is "angle addition postulate".