Question

1. use the diagram below to complete each part.

this is a 2 - page document!
unit 1: geometry basics
homework 4: angle addition postulate
name: name written per: number written date: date written

• 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: answer written ∠ebf: answer written ∠abc: answer written
e) name an angle bisector.
f) if m∠ebd = 36° and m∠dbc = 108°, find m∠ebc.
a) if m∠ebf = 117°, find m∠abe.

Explanation:

Step1: Identify vertex

The vertex of an angle is the common - point where the rays forming the angle meet. For ∠4, the vertex is point B.

Step2: Identify sides of an angle

The sides of ∠1 are rays $\overrightarrow{BA}$ and $\overrightarrow{BC}$.

Step3: Find another name for an angle

Another name for ∠5 is ∠EBD since the vertex is B and the rays are $\overrightarrow{BE}$ and $\overrightarrow{BD}$.

Step4: Classify angles

Based on the measures of angles (not given in full here for a full - classification, but generally), we can classify angles as acute (less than 90°), right (equal to 90°), obtuse (greater than 90° and less than 180°), straight (equal to 180°). Without specific measures, we can't classify precisely. For example, if we assume we had measures, we would use those values to classify.

Step5: Find angle bisector

An angle bisector divides an angle into two equal parts. Without specific angle - measure relationships indicating equal parts, we can't name an angle bisector precisely. But if we had, say, ∠ABC divided into two equal sub - angles by a ray, that ray would be the angle bisector.

Step6: Calculate angle measure

We know that ∠EBD = 36° and ∠DBC = 108°, so ∠EBC=∠EBD + ∠DBC.
$m\angle EBC=36^{\circ}+ 108^{\circ}=144^{\circ}$
We also know that ∠EBF = 117°, and ∠EBC = 144°, so ∠FBC=∠EBC - ∠EBF.
$m\angle FBC=144^{\circ}-117^{\circ}=27^{\circ}$
To find ∠ABE, we note that ∠EBF = 117° and assume that ∠ABE and ∠EBF are related. If we consider the whole angle formed around point B, we need more information about the non - overlapping parts. But if we assume that ∠ABE and ∠EBF are adjacent and make up a larger angle, and we know ∠EBF = 117°, and assume the relevant geometric relationships, we can calculate. However, if we assume that ∠ABE and ∠EBF are such that we know the whole and part relationship, and we know ∠EBF = 117°, and assume the relevant angle - addition postulate, we find that if we consider the angles around point B and the given information, we first find that ∠EBC = 144° and ∠EBF = 117°. Since ∠EBC=∠EBF + ∠FBC, we found ∠FBC = 27°. Now, if we assume that ∠ABE and ∠EBF are adjacent and we know the relationships between the angles around point B, we find that ∠ABE=∠EBF - ∠ABF. But we need to establish more relationships. If we assume that ∠ABC is a straight line (180°) and we know ∠EBC = 144°, then ∠ABE = 180°-∠EBC.
$m\angle ABE = 180^{\circ}-144^{\circ}=36^{\circ}$

Answer:

a. Vertex of ∠4: B
b. Sides of ∠1: $\overrightarrow{BA}$, $\overrightarrow{BC}$
c. Another name for ∠5: ∠EBD
d. (No specific classification given measures)
e. (No angle bisector identified without more info)
f. $m\angle EBC = 144^{\circ}$, $m\angle FBC=27^{\circ}$, $m\angle ABE = 36^{\circ}$