Triangle EAM is an isosceles triangle with base $\overline{\mathrm{ME}}.$ Line segment AE bisects $\angle \mathrm{TEM}$ and $\overline{\mathrm{AM}}$ bisects $\angle \mathrm{TME}.$ Prove that $\mathrm{\Delta MET}$ is an isosceles triangle.

Proof:

Statements	Reasons
1. $\mathrm{\Delta EAM}$ is an isosceles triangle with base $\overline{\mathrm{ME}}.$ AE bisects $\angle \mathrm{TEM}$ and $\overline{\mathrm{AM}}$ bisects $\angle \mathrm{TME}.$	1. Given
2.	2. Definition of isosceles triangle
3. $\angle \mathrm{AEM}\cong \angle \mathrm{AME}$	3.
4. $\angle \mathrm{TEA}\cong \angle \mathrm{AEM}$ $\angle \mathrm{TMA}\cong \angle \mathrm{AME}$	4.
5. $m\angle \mathrm{TEA}=m\angle \mathrm{AEM}$ $m\angle \mathrm{TMA}=m\angle \mathrm{AME}$ $m\angle \mathrm{AEM}=m\angle \mathrm{AME}$	5. Definition of congruent angles
6. $m\angle \mathrm{AEM}+m\angle \mathrm{AEM}$ $=m\angle \mathrm{AME}+m\angle \mathrm{AME}$	6.
7.	7. Substitution Property of Equality
8. $m\angle \mathrm{TEM}=m\angle \mathrm{TME}$	8.
9.	9. Definition of congruent angles
10.	10. Converse of the Isosceles Triangle Theorem
11. $\mathrm{\Delta MET}$ is an isosceles triangle.	11. Definition of isosceles triangle