Completed geometry homework on the Angle Addition Postulate, showing student work with diagrams and calculations.

Problem Analysis and Solution

The provided worksheet involves basic geometry concepts, specifically focusing on angles, angle addition postulate, and solving for unknown variables. Let's go through each problem step by step.

---

Problem 1: Use the diagram below to complete each part.

#### Given Diagram:
- A geometric figure with points \( A, B, C, D, E, F \).
- Angles labeled as \( \angle 1, \angle 2, \angle 3, \angle 4, \angle 5 \).
- Lines \( \overline{BF} \perp \overline{AC} \).

#### Questions:

##### a) Name the vertex of \( \angle 4 \).
- The vertex of an angle is the point where the two rays (sides) meet.
- From the diagram, \( \angle 4 \) has its vertex at point \( B \).
- Answer: \( B \)

##### b) Name the sides of \( \angle 1 \).
- The sides of an angle are the two rays that form it.
- \( \angle 1 \) is formed by rays \( \overrightarrow{BA} \) and \( \overrightarrow{BC} \).
- Answer: \( \overrightarrow{BA} \) and \( \overrightarrow{BC} \)

##### c) Write another name for \( \angle 5 \).
- \( \angle 5 \) can also be named using three points, where the middle point is the vertex.
- Another name for \( \angle 5 \) is \( \angle CBD \).
- Answer: \( \angle CBD \)

##### d) Classify each angle:
- \( \angle FBC \): This angle is a right angle because \( \overline{BF} \perp \overline{AC} \).
- Classification: \( 90^\circ \)
- \( \angle EBF \): This angle is also a right angle because \( \overline{BF} \perp \overline{AC} \).
- Classification: \( 90^\circ \)
- \( \angle ABC \): This angle appears to be a straight angle (formed by a straight line).
- Classification: \( 180^\circ \)

##### g) Name an angle bisector.
- An angle bisector divides an angle into two equal parts.
- From the diagram, \( \overline{BF} \) bisects \( \angle ABC \).
- Answer: \( \overline{BF} \)

##### h) If \( m\angle EBD = 36^\circ \) and \( m\angle DBC = 108^\circ \), find \( m\angle EBC \).
- Using the angle addition postulate:
\[
m\angle EBC = m\angle EBD + m\angle DBC
\]
Substituting the given values:
\[
m\angle EBC = 36^\circ + 108^\circ = 144^\circ
\]
- Answer: \( 144^\circ \)

##### i) If \( m\angle EBF = 117^\circ \), find \( m\angle ABE \).
- Since \( \overline{BF} \perp \overline{AC} \), \( \angle ABF \) is a right angle (\( 90^\circ \)).
- \( \angle ABE \) and \( \angle EBF \) are adjacent angles that together form \( \angle ABF \):
\[
m\angle ABE + m\angle EBF = 90^\circ
\]
Substituting the given value for \( m\angle EBF \):
\[
m\angle ABE + 117^\circ = 90^\circ
\]
Solving for \( m\angle ABE \):
\[
m\angle ABE = 90^\circ - 117^\circ = -27^\circ
\]
However, this result is incorrect because angles cannot have negative measures. Re-evaluating the problem, it seems there might be a misunderstanding in the interpretation. Assuming \( \angle EBF \) is part of a larger context, let’s recheck:
- If \( \angle EBF = 117^\circ \) and \( \angle ABF = 90^\circ \), then \( \angle ABE \) should be:
\[
m\angle ABE = 90^\circ - (180^\circ - 117^\circ) = 90^\circ - 63^\circ = 27^\circ
\]
Correcting the approach:
\[
m\angle ABE = 90^\circ - 63^\circ = 27^\circ
\]
- Answer: \( 27^\circ \)

---

Problem 2: If \( m\angle MKL = 83^\circ \), \( m\angle JKL = 127^\circ \), and \( m\angle JKM = (9x - 10)^\circ \), find the value of \( x \).

#### Solution:
- Using the angle addition postulate:
\[
m\angle JKL = m\angle JKM + m\angle MKL
\]
Substituting the given values:
\[
127^\circ = (9x - 10)^\circ + 83^\circ
\]
Simplify the equation:
\[
127 = 9x - 10 + 83
\]
\[
127 = 9x + 73
\]
Subtract 73 from both sides:
\[
54 = 9x
\]
Divide by 9:
\[
x = 6
\]

- Answer: \( x = 6 \)

---

Problem 3: If \( m\angle EFH = (5x + 1)^\circ \), \( m\angle HFG = 62^\circ \), and \( m\angle EFG = (18x + 11)^\circ \), find each measure.

#### Solution:
- Using the angle addition postulate:
\[
m\angle EFG = m\angle EFH + m\angle HFG
\]
Substituting the given expressions:
\[
(18x + 11) = (5x + 1) + 62
\]
Simplify the equation:
\[
18x + 11 = 5x + 63
\]
Subtract \( 5x \) from both sides:
\[
13x + 11 = 63
\]
Subtract 11 from both sides:
\[
13x = 52
\]
Divide by 13:
\[
x = 4
\]

- Now, substitute \( x = 4 \) back into the expressions for the angles:
- \( m\angle EFH = 5x + 1 = 5(4) + 1 = 20 + 1 = 21^\circ \)
- \( m\angle HFG = 62^\circ \) (given)
- \( m\angle EFG = 18x + 11 = 18(4) + 11 = 72 + 11 = 83^\circ \)

- Answers:
- \( m\angle EFH = 21^\circ \)
- \( m\angle HFG = 62^\circ \)
- \( m\angle EFG = 83^\circ \)

---

Final Answers:

1.
- a) \( B \)
- b) \( \overrightarrow{BA} \) and \( \overrightarrow{BC} \)
- c) \( \angle CBD \)
- d) \( 90^\circ, 90^\circ, 180^\circ \)
- g) \( \overline{BF} \)
- h) \( 144^\circ \)
- i) \( 27^\circ \)

2. \( x = 6 \)

3.
- \( m\angle EFH = 21^\circ \)
- \( m\angle HFG = 62^\circ \)
- \( m\angle EFG = 83^\circ \)

\boxed{x = 6}