Question

1. if hk is the perpendicular bisector of fe, which of the following statements is not true.

∠fjk and ∠ejk are right angles
fj≅ej
fk≅ek
hj≅jk

Explanation:

Step1: Recall properties of perpendicular bisector

A perpendicular bisector of a line segment is perpendicular to the segment and divides it into two equal - length parts. If $\overleftrightarrow{HK}$ is the perpendicular bisector of $\overline{FE}$, then $\angle FJK=\angle EJK = 90^{\circ}$ (by definition of perpendicular lines), and $\overline{FJ}\cong\overline{EJ}$ (by definition of a bisector). Also, by the Hypotenuse - Leg (HL) congruence theorem in right - triangles $\triangle FJK$ and $\triangle EJK$ (since $\overline{FJ}\cong\overline{EJ}$ and $\overline{JK}$ is common), $\overline{FK}\cong\overline{EK}$.

Step2: Analyze each option

• Option 1: $\angle FJK$ and $\angle EJK$ are right angles. This is true because $\overleftrightarrow{HK}$ is the perpendicular bisector of $\overline{FE}$, so $\overleftrightarrow{HK}\perp\overline{FE}$.
• Option 2: $\overline{FJ}\cong\overline{EJ}$. This is true as $\overleftrightarrow{HK}$ bisects $\overline{FE}$.
• Option 3: $\overline{FK}\cong\overline{EK}$. This is true by HL congruence of $\triangle FJK$ and $\triangle EJK$.
• Option 4: There is no information given that would imply $\overline{HJ}\cong\overline{JK}$. Just because $\overleftrightarrow{HK}$ is the perpendicular bisector of $\overline{FE}$ does not mean it bisects $\overline{HK}$ itself.

Answer:

$\overline{HJ}\cong\overline{JK}$ is not true.