We are given that $\angle MYT \cong \angle NYT$ and $\angle MTY \cong \angle NTY$. We want to prove that $\triangle RYM \cong \triangle RYN$.

1. Problem Description

We are given that $\angle MYT \cong \angle NYT$ and $\angle MTY \cong \angle NTY$. We want to prove that $\triangle RYM \cong \triangle RYN$.

2. Solution Steps

Statements | Reasons

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1. $\angle MYT \cong \angle NYT$ |

1. Given

2. $\angle MTY \cong \angle NTY$ |

2. Given

3. $\overline{YT} \cong \overline{YT}$ |

3. Reflexive Property of Congruence

4. $\triangle MTY \cong \triangle NTY$ |

4. ASA Congruence Postulate (using steps 1, 2, and 3)

5. $\overline{MY} \cong \overline{NY}$ |

5. CPCTC (Corresponding Parts of Congruent Triangles are Congruent)

6. $\angle RYM \cong \angle RYN$ |

6. Definition of perpendicularity. If $\angle MYT \cong \angle NYT$, then the line containing $\overline{YT}$ is an angle bisector of $\angle MYN$. Since the two angles are equal and sum to 180, they are right angles. If $\overline{YT}$ is perpendicular to $\overline{MN}$, then $\angle RYM$ and $\angle RYN$ are both 90 degrees and congruent.

7. $\overline{RY} \cong \overline{RY}$ |

7. Reflexive Property of Congruence

8. $\triangle RYM \cong \triangle RYN$ |

8. SAS Congruence Postulate (using steps 5, 6, and 7)

3. Final Answer

$\triangle RYM \cong \triangle RYN$