Reasons:
a. CPCTC
b. SAA Congruence Theorem
c. Definition of perpendicular segments
d. Vertical Angles Theorem
e. HL Congruence Theorem

Proof:

Statements	Reasons
1. $\overline{\mathrm{ST}}\perp \stackrel{\text{\_\_\_}}{\mathrm{TY}}$ $\overline{\mathrm{DY}}\perp \stackrel{\text{\_\_\_}}{\mathrm{TY}}$ $$\overline{\mathrm{SY}}\cong \overline{\mathrm{DT}}$$	1. Given
2. $\angle \mathrm{STY}$ and $\angle \mathrm{DYT}$ are right angles.	2.
3. $\mathrm{\Delta STY}$ and $\mathrm{\Delta DYT}$ are right triangles	3. Definition of a right triangle
4. $\stackrel{\text{\_\_\_}}{\mathrm{TY}}\cong \stackrel{\text{\_\_\_}}{\mathrm{TY}}$	4. Reflexive Property of Segment Congruence
5. $\mathrm{\Delta STY}\cong \mathrm{\Delta DYT}$	5.
6. $\angle S\cong \angle D,\overline{\mathrm{ST}}\cong \overline{\mathrm{DY}}$	6. CPCTC
7. $$\angle \mathrm{SUT}\cong \angle \mathrm{DUY}$$	7.
8. $\mathrm{\Delta STU}\cong \mathrm{\Delta DYU}$	8.
9. $\overline{\mathrm{SU}}\cong \overline{\mathrm{DU}}$	9.

Reasons:
a. Angle Addition Postulate
b. CPCTC
c. Addition Property of Equality
d. Definition of congruent angles
e. Reflexive Property of Segment Congruence

Proof:

Statements	Reasons
1. $\angle \mathrm{GEM}\cong \angle \mathrm{MOG}$ $\angle \mathrm{MEO}\cong \angle \mathrm{GOE}$	1. Given
2. $m\angle \mathrm{GEM}=m\angle \mathrm{MOG}$ $m\angle \mathrm{MEO}=m\angle \mathrm{GOE}$	2.
3. $m\angle \mathrm{GEM}+m\angle \mathrm{MEO}$ $=m\angle \mathrm{MOG}+m\angle \mathrm{GOE}$	3.
4. $m\angle \mathrm{GEO}=m\angle \mathrm{MOE}$	4.
5. $\angle \mathrm{GEO}\cong \angle \mathrm{MOE}$	5. Definition of congruent angles
6. $\overline{\mathrm{EO}}\cong \overline{\mathrm{EO}}$	6.
7. $\mathrm{\Delta GEO}\cong \mathrm{\Delta MOE}$	7. ASA Congruence Postulate
8. $\angle G\cong \angle M$	8.