Question

problem 23
given: ∠d ≅ ∠h, df ≅ hf
prove: dg ≅ he

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2. given
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4. reflexive property
5. △dgf ≅ △hef
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Explanation:

Step1: State given information

Given $\angle D\cong\angle H,\overline{DF}\cong\overline{HF}$

Step2: Identify vertical angles

$\angle DFG$ and $\angle HFE$ are vertical angles, so $\angle DFG\cong\angle HFE$ by the reflexive property of congruence for vertical angles.

Step3: Prove triangle congruence

Since we have $\angle D\cong\angle H$, $\overline{DF}\cong\overline{HF}$ and $\angle DFG\cong\angle HFE$, $\triangle DGF\cong\triangle HEF$ by the ASA congruence postulate.

Step4: Prove segment congruence

Since $\triangle DGF\cong\triangle HEF$, $\overline{DG}\cong\overline{HE}$ because corresponding parts of congruent triangles are congruent (CPCTC).

Answer:

1. $\angle D\cong\angle H,\overline{DF}\cong\overline{HF}$
2. $\angle DFG\cong\angle HFE$
3. By the Angle - Side - Angle (ASA) Congruence Postulate
4. $\overline{DG}\cong\overline{HE}$ (Corresponding parts of congruent triangles are congruent - CPCTC)