Question

1. use the diagram below to complete each part.

a) name the vertex of ∠4.
b) name the sides of ∠1.
c) write another name for ∠5.
d) classify each angle: ∠fbc, ∠ebf.
e) name an angle bisector.
if m∠ebd = 36° and m∠dbc = 108°, find m∠ebc.
f) if m∠ebf = 117°, find m∠abe, m∠ebc.
g) (overline{bf}perpoverline{ac})

Explanation:

Step1: Identify vertex of ∠4

The vertex of an angle is the common endpoint of the rays that form the angle. For ∠4, the vertex is point B.

Step2: Identify sides of ∠1

The sides of ∠1 are the two rays that form the angle. For ∠1, the sides are rays $\overrightarrow{BD}$ and $\overrightarrow{BC}$.

Step3: Find another name for ∠5

Another name for ∠5 can be ∠DBE since the angle can be named by its vertex (B) with the rays $\overrightarrow{BD}$ and $\overrightarrow{BE}$ forming it.

Step4: Classify ∠EBF

An acute angle is less than 90°. Since no information indicates otherwise and we assume based on typical angle - naming problems, if we consider a normal range of angles, ∠EBF is an acute angle.

Step5: Name an angle bisector

Given $\overrightarrow{BF}\perp\overrightarrow{AC}$, if we assume that $\overrightarrow{BF}$ divides some angle related to the intersection of $\overrightarrow{BF}$ and $\overrightarrow{AC}$ equally, an angle bisector could be $\overrightarrow{BF}$ (for example, if it bisects a right - angle formed at the intersection with $\overrightarrow{AC}$).

Step6: Find $m\angle EBC$

If $\overrightarrow{BD}$ is an angle bisector, and $m\angle EBD = 36^{\circ}$, then $m\angle EBC=2\times m\angle EBD$. So $m\angle EBC = 2\times36^{\circ}=72^{\circ}$.

Step7: Find $m\angle ABE$

We know that $m\angle EBF = 117^{\circ}$ and $\overrightarrow{BF}\perp\overrightarrow{AC}$, so the right - angle formed by $\overrightarrow{BF}$ and $\overrightarrow{AC}$ is $90^{\circ}$. Let's assume that $\angle ABE+\angle EBF = 180^{\circ}$ (a straight - line angle). Then $m\angle ABE=180^{\circ}-m\angle EBF$. So $m\angle ABE = 180^{\circ}-117^{\circ}=63^{\circ}$.

Answer:

a) Vertex of ∠4: B
b) Sides of ∠1: $\overrightarrow{BD}$ and $\overrightarrow{BC}$
c) Another name for ∠5: ∠DBE
d) Classification of ∠EBF: Acute angle
e) Angle bisector: $\overrightarrow{BF}$
f) $m\angle EBC = 72^{\circ}$
g) $m\angle ABE = 63^{\circ}$