given: $ \overline { f g } \cong \overline { f h } $, $ \overline { f k } $ bisects $ \angle h f g $. prove: $ \triangle g j k \cong \triangle h j k $ 6. 7. $ \angle g j f $ and $ \angle g j k $ are supplementary; $ \angle h j f $ and $ \angle h j k $ are supplementary. 8. $ \angle g j k \cong \angle h j k $ 6. corresponding parts of congruent tri - angles are congruent (c.p.c.t.c.) 7. if two angles are a linear pair, then they are supplementary. 8.

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# Explanation:
## Step1: Use the definition of angle bisector
Since $ \overline{FK} $ bisects $ \angle HFG $, then $ \angle GFK=\angle HFK $.
## Step2: Use the Side - Angle - Side (SAS) congruence criterion
We know that $ \overline{FG}\cong\overline{FH} $ (given), $ \angle GFK = \angle HFK $ (from step 1), and $ \overline{FK}\cong\overline{FK} $ (common side). So, $ \triangle FGK\cong\triangle FHK $ by SAS.
## Step3: Use the Corresponding Parts of Congruent Triangles are Congruent (CPCTC)
Since $ \triangle FGK\cong\triangle FHK $, then $ \overline{GK}\cong\overline{HK} $ and $ \angle FGK=\angle FHK $.
## Step4: Consider $ \angle GJF $ and $ \angle GJK $ are supplementary; $ \angle HJF $ and $ \angle HJK $ are supplementary (given)
Let $ \angle GJF = x$, then $ \angle GJK=180 - x$. Let $ \angle HJF = y$, then $ \angle HJK = 180 - y$.
## Step5: Use the fact that $ \angle FGK=\angle FHK $ (from step 3) and angles in triangles
In $ \triangle GJF$ and $ \triangle HJF$, we can find relations for angles. But since $ \triangle FGK\cong\triangle FHK $, and considering the linear - pair supplementary angles.
## Step6: Use the Side - Side - Side (SSS) or Side - Angle - Side (SAS) for $ \triangle GJK$ and $ \triangle HJK$
We have $ \overline{GK}\cong\overline{HK} $ (from step 3), $ \overline{JK}\cong\overline{JK} $ (common side), and $ \angle GJK\cong\angle HJK $ (from step 4 and angle relations). So, $ \triangle GJK\cong\triangle HJK $ by SAS.

# Answer:
$ \triangle GJK\cong\triangle HJK $

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QUESTION IMAGE

Question

Explanation:

Step1: Use the definition of angle bisector

Step2: Use the Side - Angle - Side (SAS) congruence criterion

Step3: Use the Corresponding Parts of Congruent Triangles are Congruent (CPCTC)

Step4: Consider \( \angle GJF \) and \( \angle GJK \) are supplementary; \( \angle HJF \) and \( \angle HJK \) are supplementary (given)

Step5: Use the fact that \( \angle FGK=\angle FHK \) (from step 3) and angles in triangles

Step6: Use the Side - Side - Side (SSS) or Side - Angle - Side (SAS) for \( \triangle GJK\) and \( \triangle HJK\)

Answer: