Question

given: m∠aeb = 45° ∠aec is a right angle. prove: eb bisects ∠aec. proof: we are given that m∠aeb = 45° and ∠aec is a right angle. the measure of ∠aec is 90° by the definition of a right angle. applying the segment addition property gives m∠aeb + m∠bec = 90°. the substitution gives m∠bec = 45°, so ∠aeb and ∠bec have the same measure. eb divides ∠aec into two congruent angles, so eb bisects ∠aec.

Explanation:

Step1: Recall right - angle measure

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

Step3: Solve for $m\angle BEC$

Using the subtraction property, we get $m\angle BEC=90^{\circ}- 45^{\circ}=45^{\circ}$.

Step4: Define angle - bisector

Since $m\angle AEB = 45^{\circ}$ and $m\angle BEC = 45^{\circ}$, $\overrightarrow{EB}$ divides $\angle AEC$ into two congruent angles. By the definition of an angle - bisector, $\overrightarrow{EB}$ bisects $\angle AEC$.

Answer:

The missing property in the proof is the angle addition postulate.