Question

proof: any point on the perpendicular bisector of a line segment is equidistant from the endpoints of the line segment.
given
a segment and its perpendicular bisector
perpendicular bisector (definition)
midpoint (definition and theorem)
________________
perpendicular (definition)
right angles (all right angles are congruent)
________________
reflexive property of congruence
________________
$\triangle apm \cong \triangle bpm$ by ________
$\overline{ap} \cong \overline{bp}$ by _____________
definition of _____________
$ap = bp$

Explanation:

Step1: Fill midpoint conclusion

From midpoint definition, $\overline{AM} \cong \overline{BM}$

Step2: Fill perpendicular angle conclusion

Right angles are congruent, so $\angle AMP \cong \angle BMP$

Step3: Fill reflexive property statement

Reflexive property gives $\overline{PM} \cong \overline{PM}$

Step4: Fill triangle congruence rule

Use SAS: $\triangle APM \cong \triangle BPM$ by SAS Congruence

Step5: Fill congruent segments reason

Corresponding parts of congruent triangles are congruent (CPCTC): $\overline{AP} \cong \overline{BP}$ by CPCTC

Step6: Fill final definition

Definition of congruent segments: $AP = BP$ by Definition of Congruent Segments

Answer:

1. Midpoint (definition and theorem) blank: $\boldsymbol{\overline{AM} \cong \overline{BM}}$
2. Perpendicular (definition) blank: $\boldsymbol{\angle AMP \cong \angle BMP}$
3. Reflexive Property of Congruence blank: $\boldsymbol{\overline{PM} \cong \overline{PM}}$
4. $\triangle APM \cong \triangle BPM$ blank: $\boldsymbol{SAS \ Congruence}$
5. $\overline{AP} \cong \overline{BP}$ blank: $\boldsymbol{CPCTC}$
6. Definition of blank: $\boldsymbol{Congruent \ Segments}$