Question

1. use the diagram below to complete each part.

• bf⊥ac

a) name the vertex of ∠4.
b) name the sides of ∠1.
c) write another name for ∠5.
d) classify each angle:
∠fbc: ∠ebf: ∠abc:
g) name an angle bisector.
h) if m∠ebd = 36° and m∠dbc = 108°, find m∠ebc.
i) if m∠ebf = 117°, find m∠abe.

Explanation:

Step1: Identify vertex of an angle

The vertex of an angle is the common endpoint of the rays that form the angle. For $\angle4$, the vertex is $B$.

Step2: Name sides of an angle

The sides of $\angle1$ are the two rays that form the angle. The sides of $\angle1$ are $\overrightarrow{BD}$ and $\overrightarrow{BC}$.

Step3: Alternative angle naming

Another name for $\angle5$ is $\angle DBE$.

Step4: Classify angles

• $\angle FBC$: Since $\overrightarrow{BF}\perp\overrightarrow{AC}$, $\angle FBC = 90^{\circ}$, so it is a right - angle.
• $\angle EBF$: It is greater than $90^{\circ}$ and less than $180^{\circ}$, so it is an obtuse - angle.
• $\angle ABC$: It is less than $90^{\circ}$, so it is an acute - angle.

Step5: Identify angle bisector

If an angle is divided into two equal angles, the line dividing it is an angle bisector. If we assume $\overrightarrow{BD}$ divides $\angle EBC$ into two equal parts (not given explicitly but a possible candidate from the diagram), $\overrightarrow{BD}$ could be an angle bisector.

Step6: Find angle measure by addition

For $m\angle EBC$, since $\angle EBC=\angle EBD+\angle DBC$, and $m\angle EBD = 36^{\circ}$ and $m\angle DBC = 108^{\circ}$, then $m\angle EBC=36^{\circ}+ 108^{\circ}=144^{\circ}$.

Step7: Find angle measure by subtraction

Since $\angle EBF = 117^{\circ}$ and $\angle ABF = 90^{\circ}$ (because $\overrightarrow{BF}\perp\overrightarrow{AC}$), then $m\angle ABE=m\angle EBF - m\angle ABF=117^{\circ}-90^{\circ}=27^{\circ}$.

Answer:

a) $B$
b) $\overrightarrow{BD}$ and $\overrightarrow{BC}$
c) $\angle DBE$
d) $\angle FBC$: right - angle; $\angle EBF$: obtuse - angle; $\angle ABC$: acute - angle
g) $\overrightarrow{BD}$ (possible)
h) $144^{\circ}$
i) $27^{\circ}$