Question

1. write a two column proof of the perpendicular bisector theorem

given: $overline{cd}$ is the perpendicular bisector of $overline{ab}$.
prove: c is equidistant from a and b.

Explanation:

Step1: State given info

$\overline{CD}$ is perpendicular bisector of $\overline{AB}$, so $\overline{AE} = \overline{EB}$, $\angle CEA = \angle CEB = 90^\circ$

Step2: Identify shared side

$\overline{CE} = \overline{CE}$ (Reflexive Property)

Step3: Apply SAS congruence

$\triangle CEA \cong \triangle CEB$ (SAS: $\overline{AE}=\overline{EB}$, $\angle CEA=\angle CEB$, $\overline{CE}=\overline{CE}$)

Step4: Corresponding parts are equal

$\overline{CA} = \overline{CB}$ (CPCTC: Corresponding Parts of Congruent Triangles are Congruent)

Answer:

Two-column proof:

Statements	Reasons
2. $\overline{AE} = \overline{EB}$, $\angle CEA = \angle CEB = 90^\circ$	2. Definition of perpendicular bisector
3. $\overline{CE} = \overline{CE}$	3. Reflexive Property of Congruence
4. $\triangle CEA \cong \triangle CEB$	4. SAS Congruence Postulate
5. $\overline{CA} = \overline{CB}$	5. CPCTC
6. $C$ is equidistant from $A$ and $B$	6. Definition of equidistant