Question

prove #1
given: ∠m≅∠h, ∠mat≅∠hta
prove: △mat≅△hta
statements

prove #2
given: uf and vh bisect each other
prove: △uvg≅△fhg
statement
reasons

Explanation:

Proof 1

Step1: Identify given congruent parts

Given $\angle M\cong\angle H$ and $\overline{MA}\cong\overline{HA}$. Also, $\angle MAT\cong\angle HAT$ because they are the same angle.

Step2: Apply congruence criterion

By the Angle - Side - Angle (ASA) congruence criterion, since we have two pairs of congruent angles and the included side congruent between $\triangle MAT$ and $\triangle HAT$, we can say $\triangle MAT\cong\triangle HAT$.

Proof 2

Step1: Use the property of bisecting lines

Since $\overline{UF}$ and $\overline{VH}$ bisect each other, let the intersection point be $G$. Then $\overline{UG}\cong\overline{FG}$ and $\overline{VG}\cong\overline{HG}$ (by the definition of bisection).

Step2: Identify vertical angles

$\angle UGV\cong\angle FGH$ because they are vertical angles.

Step3: Apply congruence criterion

By the Side - Angle - Side (SAS) congruence criterion, since we have two pairs of congruent sides and the included angle congruent between $\triangle UVG$ and $\triangle FHG$, we can say $\triangle UVG\cong\triangle FHG$.

Answer:

Proof 1: $\triangle MAT\cong\triangle HAT$ by ASA.
Proof 2: $\triangle UVG\cong\triangle FHG$ by SAS.