Reasons:
a. Definition of linear pair
b. Linear Pair Postulate
c. Definition of Isosceles Triangle
d. Converse of the Isosceles Triangle Theorem
e. Supplement Theorem

Proof:

Statements	Reasons
1. $\angle 1\cong \angle 2$	1. Given
2. $\angle 1$ and $\angle 3$ form a linear pair. $\angle 2$ and $\angle 4$ form a linear pair.	2.
3. $\angle 1$ and $\angle 3$ are supplementary angles. $\angle 2$ and $\angle 4$ are supplementary angles.	3.
4. $\angle 3\cong \angle 4$	4.
5. $\overline{\mathrm{AT}}\cong \overline{\mathrm{AP}}$	5.
6. $\mathrm{\Delta TAP}$ is an isosceles triangle.	6.

Reasons:
a. Definition of a linear pair
b. Isosceles Triangle Theorem
c. Definition of median
d. SAS Congruence Postulate
e. If two angles are supplementary and congruent, then each is a right angle.
f. Statements 3 and 10, and definition of perpendicular bisector

Proof:

Statements	Reasons
1. $\overline{\mathrm{AC}}\cong \overline{\mathrm{BC}}$ Median $\overline{\mathrm{CD}}$ of $\mathrm{\Delta ABC}$	1. Given
2. D is the midpoint of $\stackrel{\text{\_\_}}{\mathrm{AB}}.$	2.
3. $\overline{\mathrm{AD}}\cong \overline{\mathrm{DB}}$	3. Definition of midpoint
4. $\angle A\cong \angle B$	4.
5. $\mathrm{\Delta CAD}\cong \mathrm{\Delta CBD}$	5.
6. $\angle \mathrm{CDA}\cong \angle \mathrm{CDB}$	6. CPCTC
7. $\angle \mathrm{CDA}$ and $\angle \mathrm{CDB}$ form a linear pair.	7.
8. $\angle \mathrm{CDA}$ and $\angle \mathrm{CDB}$ are supplementary.	8. Linear Pair Postulate
9. $\angle \mathrm{CDA}$ and $\angle \mathrm{CDB}$ are right angles.	9.
10. $\overline{\mathrm{CD}}$ is perpendicular to $\stackrel{\text{\_\_}}{\mathrm{AB}}.$	10. Definition of perpendicular segments
11. $\overline{\mathrm{CD}}$ is the perpendicular bisector of $\stackrel{\text{\_\_}}{\mathrm{AB}}.$	11.