1. **Problem Statement:** Prove that triangles $\triangle HGI$ and $\triangle CID$ are congruent given: - $\angle HGI \cong \angle CID$ - $\angle CDI$ is a right angle - $HI$ is the perpendicular bisector of $GD$ 2. **Identify Known Information and What to Prove:** - We know two angles are congruent. - $HI$ being the perpendicular bisector means $HI \perp GD$ and $GI = ID$. - We want to prove $\triangle HGI \cong \triangle CID$. 3. **Congruence Criteria:** - Use the Angle-Side-Angle (ASA) criterion: two angles and the included side are congruent. 4. **Proof Steps:** - Since $HI$ is the perpendicular bisector of $GD$, $GI = ID$ (by definition of bisector). - $\angle HGI \cong \angle CID$ (given). - $\angle CDI$ is a right angle, so $\angle HIG$ is also right because $HI \perp GD$. - Therefore, $\angle HIG \cong \angle CDI$ (both right angles). - By ASA, $\triangle HGI \cong \triangle CID$. 5. **Summary:** - We used the given angle congruences and the property of the perpendicular bisector to establish two angles and the included side congruent. - Hence, $\triangle HGI \cong \triangle CID$ by ASA. 2. **Problem Statement:** Prove that triangles $\triangle MAT$ and $\triangle HTA$ are congruent given: - $\angle M \cong \angle H$ - $\angle MAT \cong \angle HTA$ 3. **Congruence Criteria:** - Use the Angle-Angle-Side (AAS) or Angle-Side-Angle (ASA) criterion. 4. **Proof Steps:** - $\angle M \cong \angle H$ (given). - $\angle MAT \cong \angle HTA$ (given). - Side $AT$ is common to both triangles. - By ASA, $\triangle MAT \cong \triangle HTA$. 5. **Summary:** - Two angles and the included side are congruent. - Therefore, $\triangle MAT \cong \triangle HTA$ by ASA. **Note:** To mark the picture with red lines, arcs, and labels, use the following: - Draw arcs at congruent angles. - Mark equal sides with tick marks. - Label right angles with a small square. - Highlight the perpendicular bisector with a red line. This completes the proofs for both triangle congruences.