Question

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triangle fgh is the image of isosceles triangle feh after a reflection across line hf. which of the following statements must be true?
efhg is a rectangle.
diagonal fh bisects angles efg and ehg.
diagonal fh is perpendicular to side fe.
angle efh is congruent to angle feh.

Explanation:

Step1: Recall properties of reflection

A reflection is a rigid - motion that preserves angle measures and side lengths. Since $\triangle FGH$ is the image of $\triangle FEH$ after a reflection across line $HF$, line $HF$ is the line of symmetry.

Step2: Analyze angle - bisecting property

For a figure that is symmetric about a line (in this case, the quadrilateral $EFGH$ is symmetric about line $HF$), the line of symmetry (diagonal $HF$) bisects the angles formed by the non - parallel sides at the vertices it passes through. So, diagonal $FH$ bisects angles $EFG$ and $EHG$.

Step3: Analyze other options

• Option 1: Just because $\triangle FEH$ is isosceles and $\triangle FGH$ is its reflection across $HF$, we cannot conclude that $EFGH$ is a rectangle. There is no information about right - angles.
• Option 3: There is no information to suggest that $FH$ is perpendicular to $FE$. In an isosceles triangle $\triangle FEH$ with $FE = HE$, $FH$ is not necessarily perpendicular to $FE$.
• Option 4: In $\triangle FEH$, $\angle EFH$ and $\angle FEH$ are not congruent in general. $\angle EFH=\angle HFG$ due to reflection, but there is no reason for $\angle EFH$ to be congruent to $\angle FEH$.

Answer:

Diagonal FH bisects angles EFG and EHG.