Answer key for identifying basic geometric shapes.

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.

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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 \( \angle 1, \angle 2, \angle 3, \angle 4, \angle 5 \).
- Line segments and rays are marked, and some angles are given specific measures.
- Note: \( \overline{BF} \perp \overline{AC} \), meaning \( \overline{BF} \) is perpendicular to \( \overline{AC} \).

#### Questions and Solutions:

##### a) Name the vertex of \( \angle 4 \).
- The vertex of an angle is the point where the two rays 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 spans from \( \overrightarrow{AB} \) to \( \overrightarrow{BC} \) and appears to be a straight angle.
- Classification: \( 180^\circ \)

##### g) Name an angle bisector.
- An angle bisector divides an angle into two equal parts.
- From the diagram, \( \overline{BF} \) appears to bisect \( \angle ABC \).
- 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 (\( 90^\circ \)).
- \( \angle EBF \) is composed of \( \angle ABE \) and \( \angle ABF \):
\[
m\angle EBF = m\angle ABE + m\angle ABF
\]
Substituting the known values:
\[
117^\circ = m\angle ABE + 90^\circ
\]
Solving for \( m\angle ABE \):
\[
m\angle ABE = 117^\circ - 90^\circ = 27^\circ
\]
- Answer: \( 27^\circ \)

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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:
- From the diagram, \( \angle JKL \) is composed of \( \angle JKM \) and \( \angle MKL \):
\[
m\angle JKL = m\angle JKM + m\angle MKL
\]
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: \( x = 6 \)

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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:
- From the diagram, \( \angle EFG \) is composed of \( \angle EFH \) and \( \angle HFG \):
\[
m\angle EFG = m\angle EFH + m\angle HFG
\]
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 each angle:
- \( m\angle EFH = 5x + 1 = 5(4) + 1 = 20 + 1 = 21^\circ \)
- \( m\angle HFG = 62^\circ \) (given)
- \( m\angle EFG = 18x + 11 = 18(4) + 11 = 72 + 11 = 83^\circ \)

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

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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 Final Answer:
\[
\boxed{x = 6}
\]