Question

1. the diagram below shows the construction of the bisector of ∠abc. which statement is not true? (1) m∠ebf = 1/2 m∠abc (2) m∠dbf = 1/2 m∠abc (3) m∠ebf = m∠abf (4) m∠dbf = m∠ebf

Explanation:

Step1: Recall angle - bisector property

An angle - bisector of an angle divides the angle into two equal angles. If $\overrightarrow{BF}$ is the bisector of $\angle ABC$, then $m\angle ABF=m\angle CBF=\frac{1}{2}m\angle ABC$. Also, $m\angle ABF + m\angle CBF=m\angle ABC$.

Step2: Analyze each option

• Option (1): $m\angle EBF=\frac{1}{2}m\angle ABC$. Since $\overrightarrow{BF}$ is the angle - bisector of $\angle ABC$, this can be true if $\overrightarrow{BE}$ is a ray such that $\angle EBF$ is one of the two equal angles formed by the bisector.
• Option (2): $m\angle DBF=\frac{1}{2}m\angle ABC$. This can be true if $\overrightarrow{BD}$ is a ray such that $\angle DBF$ is one of the two equal angles formed by the bisector.
• Option (3): $m\angle EBF=m\angle ABC$. This is not possible because if $\overrightarrow{BF}$ is the bisector of $\angle ABC$, then the non - overlapping sub - angles formed by other rays with $\overrightarrow{BF}$ within $\angle ABC$ will be less than or equal to $\frac{1}{2}m\angle ABC$.
• Option (4): $m\angle DBF = m\angle EBF$. This can be true if $\overrightarrow{BD}$ and $\overrightarrow{BE}$ coincide or are symmetrically placed with respect to $\overrightarrow{BF}$.

Answer:

(3)