given: $\overline{hf} \parallel \overline{jk}$; $\overline{hg} \cong \overline{jg}$ prove: $\triangle fhg \cong \triangle kjg$ to prove that the triangles are congruent by asa, which statement and reason could be used as part of the proof? $\bigcirc$ $\angle fgh \cong \angle kgj$ because vertical angles are congruent. $\bigcirc$ $\angle jkg \cong \angle hfg$ because vertical angles are congruent. $\bigcirc$ $\angle fhg \cong \angle jkg$ because right angles are congruent. $\bigcirc$ $\angle hfg \cong \angle kjg$ because alternate interior angles are congruent.

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Accepted Answer

# Answer:
D. $ \angle HFG\cong\angle KJG $ because alternate interior angles are congruent.

# Explanation:
## Step1: Recall the properties of parallel lines
When two parallel lines ($ \overline{HF}\parallel\overline{JK} $) are cut by a transversal ($ \overline{FGK} $), alternate - interior angles are congruent.
## Step2: Identify the alternate - interior angles
For the parallel lines $ \overline{HF}$ and $ \overline{JK}$ and transversal $ \overline{FGK}$, $ \angle HFG$ and $ \angle KJG$ are alternate - interior angles.
## Step3: Check the ASA (Angle - Side - Angle) criterion
We know that $ \overline{HG}\cong\overline{JG}$ (given). If we can show two angles and the included side are congruent. The included side is $ \overline{HG}$ and $ \overline{JG}$, and if $ \angle HFG\cong\angle KJG$ (from alternate - interior angles due to $ \overline{HF}\parallel\overline{JK}$) and $ \angle FGH\cong\angle KGJ$ (vertical angles are congruent), we can use ASA.
- Option A: $ \angle FGH\cong\angle KGJ$ is about vertical angles, but this option alone does not help in getting the correct pair of angles for the two triangles $ \triangle FHG$ and $ \triangle KJG$ in the context of the parallel - line - related angles for ASA.
- Option B: $ \angle JKG$ and $ \angle HFG$ are not vertical angles.
- Option C: There is no information given that these are right angles.
- Option D: Since $ \overline{HF}\parallel\overline{JK}$, by the alternate - interior angles theorem, $ \angle HFG\cong\angle KJG$. This gives us one pair of congruent angles (along with $ \overline{HG}\cong\overline{JG}$ and $ \angle FGH\cong\angle KGJ$ (vertical angles)) to apply the ASA congruence criterion for $ \triangle FHG$ and $ \triangle KJG$.

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QUESTION IMAGE

Question

Explanation:

Step1: Recall the properties of parallel lines

Step2: Identify the alternate - interior angles

Step3: Check the ASA (Angle - Side - Angle) criterion

Answer: