Question

use the diagram to answer the question.
line segment $overline{cd}$ is the perpendicular bisector of line segment $overline{ab}$. which statement shows why $overline{ae}=overline{be}$?
a $mangle efa = mangle efb=90^{circ}$ by definition of perpendicular so $angle efa=angle efb$ $overline{af}=overline{bf}$ by the definition of mid - point $\triangle eafcong\triangle ebf$ by hypotenuse - leg $overline{ae}=overline{be}$ because they are corresponding parts of congruent triangles
b $mangle efa + mangle efb = 180^{circ}$ so $\triangle aeb$ must be isosceles $overline{ae}=overline{be}$ because the legs of isosceles triangles are congruent
c $mangle efa = mangle efb = 90^{circ}$ by definition of perpendicular so $angle efa=angle efb$ $overline{af}=overline{bf}$ by the definition of mid - point $overline{ef}=overline{ef}$ by the reflexive property

Explanation:

Step1: Recall perpendicular bisector properties

Since $\overline{CD}$ is the perpendicular bisector of $\overline{AB}$, by the definition of perpendicular, $m\angle EFA = m\angle EFB=90^{\circ}$, so $\angle EFA\cong\angle EFB$. Also, by the definition of a bisector, $AF = BF$.

Step2: Use reflexive property

$\overline{EF}=\overline{EF}$ by the reflexive property.

Step3: Prove triangle - congruence

In $\triangle EAF$ and $\triangle EBF$, we have $\angle EFA\cong\angle EFB$, $AF = BF$, and $\overline{EF}=\overline{EF}$. So, $\triangle EAF\cong\triangle EBF$ by the Side - Angle - Side (SAS) congruence criterion.

Step4: Use corresponding parts of congruent triangles

Since $\triangle EAF\cong\triangle EBF$, then $\overline{AE}=\overline{BE}$ because they are corresponding parts of congruent triangles.
The correct statement is:
A. $m\angle EFA = m\angle EFB = 90^{\circ}$ by definition of perpendicular so $\angle EFA=\angle EFB$, $\overline{AF}=\overline{BF}$ by the definition of mid - point, $\triangle EAF\cong\triangle EBF$ by Hypotenuse - Leg, $\overline{AE}=\overline{BE}$ because they are corresponding parts of congruent triangles.

Answer:

A. $m\angle EFA = m\angle EFB = 90^{\circ}$ by definition of perpendicular so $\angle EFA=\angle EFB$, $\overline{AF}=\overline{BF}$ by the definition of mid - point, $\triangle EAF\cong\triangle EBF$ by Hypotenuse - Leg, $\overline{AE}=\overline{BE}$ because they are corresponding parts of congruent triangles