Question

mark the picture state why the two triangles are congruent, then prove it!
proof #1
given: ∠m ≅ ∠h
∠mat ≅ ∠hta
prove: △mat ≅ △hta
statements reasons

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

Explanation:

Proof #1

Step1: Identify given information

Given $\angle M\cong\angle H$, $\angle MAT\cong\angle HTA$

Step2: Identify common side

$AT = AT$ (Reflexive property of congruence)

Step3: Apply congruence criterion

By the Angle - Angle - Side (AAS) congruence criterion, $\triangle MAT\cong\triangle HTA$ since we have two pairs of congruent angles and a non - included side congruent.

Proof #2

Step1: Use the definition of bisect

Since $UF$ and $VH$ bisect each other, we have $UG = FG$ and $VG=HG$ (Definition of bisecting segments)

Step2: Identify vertical angles

$\angle UGV\cong\angle FGH$ (Vertical angles are congruent)

Step3: Apply congruence criterion

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

Answer:

Proof #1: $\triangle MAT\cong\triangle HTA$ by AAS. Proof #2: $\triangle UVG\cong\triangle FHG$ by SAS.