Question

1. use the diagram below to complete each part.

this is a 2 - page document!
a) name the vertex of \\( \angle 4 \\).
b) name the sides of \\( \angle 1 \\).
c) write another name for \\( \angle 5 \\).
d) classify each angle:
\\( \angle fbc: \\) \\( \angle ebf: \\) \\( \angle abc: \\)
e) name an angle bisector.
f) if \\( m\angle ebd = 36 ^ { \circ } \\) and \\( m\angle dbc = 108 ^ { \circ } \\), find \\( m\angle ebc \\).
g) if \\( m\angle ebf = 117 ^ { \circ } \\), find \\( m\angle abe \\).

Explanation:

Step1: Vertex of angle

The vertex is the common point of the two rays forming the angle. For \(∠4\), the vertex is \(B\).

Step2: Sides of angle

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

Step3: Another name for angle

\(∠5\) can also be named \(∠EBD\) as it is formed by rays \(\overrightarrow{BE}\) and \(\overrightarrow{BD}\)

Step4: Classifying angles

• \(∠FBC\): Since \(\overrightarrow{BF}\perp\overrightarrow{AC}\), \(∠FBC = 90^{\circ}\), so it is a right angle.
• \(∠EBF\): \(∠EBF> 90^{\circ}\) (as \(m∠EBF = 117^{\circ}\) in part g, but generally from the diagram's structure), so it is an obtuse angle.
• \(∠ABC\): \(∠ABC = ∠1+∠2\). Since \(\overrightarrow{BF}\perp\overrightarrow{AC}\), \(∠ABC= 180^{\circ}\), so it is a straight angle.

Step5: Angle bisector

\(\overrightarrow{BE}\) bisects \(∠ABD\) (as \(∠4 = ∠5\) from the diagram's marking), so \(\overrightarrow{BE}\) is an angle bisector.

Step6: Sum of angles

For \(m∠EBC\), since \(∠EBC=∠EBD + ∠DBC\). Given \(m∠EBD = 36^{\circ}\) and \(m∠DBC = 108^{\circ}\), then \(m∠EBC=36^{\circ}+ 108^{\circ}=144^{\circ}\)

Step7: Complementary angles

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

Answer:

a) \(B\)
b) \(\overrightarrow{BC}\) and \(\overrightarrow{BD}\)
c) \(∠EBD\)
d) \(∠FBC\): right angle; \(∠EBF\): obtuse angle; \(∠ABC\): straight angle
e) \(\overrightarrow{BE}\)
f) \(144^{\circ}\)
g) \(27^{\circ}\)