Question

triangle hef is the image of triangle hgf after a reflection across line fh. select all statements that must be true. (hint: there are 3!) segment eh is congruent to segment fg. triangle efh is congruent to triangle gfh. angle hfe is congruent to angle fhg. triangle fgh is congruent to triangle feh. angle efg is congruent to angle ehg. segment gh is congruent to segment eh.

Explanation:

Step1: Recall reflection properties

A reflection is a rigid - motion. Rigid motions preserve side - lengths and angle - measures. When $\triangle HEF$ is the image of $\triangle HGF$ after a reflection across line $FH$, corresponding parts of the pre - image and image are congruent.

Step2: Identify congruent triangles

By the definition of reflection, $\triangle EHF\cong\triangle GHF$. This is because a reflection is an isometry (distance - preserving transformation), and the two triangles share side $FH$. So, $\triangle FGH\cong\triangle FEH$ (congruence is symmetric).

Step3: Identify congruent sides and angles

Corresponding sides and angles of congruent triangles are congruent. Since $\triangle EHF\cong\triangle GHF$, $\angle HFE\cong\angle FHG$ (corresponding angles of congruent triangles), and $GH\cong EH$ (corresponding sides of congruent triangles).

Answer:

• Triangle $EFH$ is congruent to triangle $GFH$.
• Triangle $FGH$ is congruent to triangle $FEH$.
• Angle $HFE$ is congruent to angle $FHG$.
• Segment $GH$ is congruent to segment $EH$.