Question

1. if de is the perpendicular bisector of ac, which of the following statements is not true? ∠abd≅∠cbd, bd≅be, ab≅bc, ad≅cd

Explanation:

Step1: Recall properties of perpendicular bisector

A perpendicular bisector of a line - segment divides the line - segment into two equal parts and is perpendicular to it. Since $\overleftrightarrow{DE}$ is the perpendicular bisector of $\overline{AC}$, we have $AB = BC$ (by the definition of a bisector) and $\angle ABD=\angle CBD = 90^{\circ}$ (by the definition of perpendicular). Also, by the Hypotenuse - Leg (HL) congruence criterion in right - triangles $\triangle ABD$ and $\triangle CBD$ (where $BD$ is common and $AB = BC$), $\triangle ABD\cong\triangle CBD$, so $AD = CD$.

Step2: Analyze each option

• Option 1: $\angle ABD\cong\angle CBD$. Since $DE\perp AC$, $\angle ABD = \angle CBD=90^{\circ}$, so this statement is true.
• Option 2: $\overline{BD}\cong\overline{BE}$. There is no information or reason to suggest that $BD$ and $BE$ are congruent. Just because $DE$ is the perpendicular bisector of $AC$ does not imply any relationship between $BD$ and $BE$. This statement is not true.
• Option 3: $\overline{AB}\cong\overline{BC}$. By the definition of a bisector, if $DE$ bisects $AC$ at $B$, then $AB = BC$, so this statement is true.
• Option 4: $\overline{AD}\cong\overline{CD}$. Using the HL congruence of $\triangle ABD$ and $\triangle CBD$ (right - triangles with $BD$ common and $AB = BC$), we can conclude that $AD = CD$, so this statement is true.

Answer:

$\overline{BD}\cong\overline{BE}$