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

• bf ⊥ ac

Explanation:

Step1: Identify vertex of ∠4

The vertex of an angle is the common endpoint of its sides. For ∠4, the vertex is point B.

Step2: Name sides of ∠1

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

Step3: Find another name for ∠5

An angle can be named by the vertex alone if there is no confusion, or by three - point naming. Another name for ∠5 is ∠EBD.

Step4: Classify ∠FBC

Since $\overline{BF}\perp\overline{AC}$, ∠FBC is a right angle (measures 90°).

Step5: Classify ∠EBF

∠EBF is an obtuse angle as it measures 117° (given in part g), and an obtuse angle is an angle with measure greater than 90° and less than 180°.

Step6: Classify ∠ABC

∠ABC is a straight angle. Since $\overline{BF}\perp\overline{AC}$, ∠ABF = ∠FBC=90°, and ∠ABC=∠ABF + ∠FBC = 180°.

Step7: Name an angle bisector

Since ∠4 = ∠5, $\overrightarrow{BD}$ is an angle - bisector of ∠EBC.

Step8: Find m∠EBC

By the angle - addition postulate, if ∠EBD and ∠DBC are adjacent angles, then m∠EBC=m∠EBD + m∠DBC. Given m∠EBD = 36° and m∠DBC = 108°, so m∠EBC=36°+ 108° = 144°.

Step9: Find m∠ABE

Since ∠EBF = 117° and ∠ABF = 90° (because $\overline{BF}\perp\overline{AC}$), then m∠ABE=m∠EBF - m∠ABF. So m∠ABE=117° - 90° = 27°.

Answer:

a) B
b) $\overrightarrow{BA}$, $\overrightarrow{BC}$
c) ∠EBD
d) ∠FBC: right angle; ∠EBF: obtuse angle; ∠ABC: straight angle
e) $\overrightarrow{BD}$
f) 144°
g) 27°