Question

part a
indicate whether each statement is true or false.

	true	false
the measure of $\angle abg$ is always equal to the measure of $\angle fbe$.		\checkmark
the sum of the measures of $\angle gbf$, $\angle fbe$, and $\angle dbc$ is always equal to $180^\circ$.		\checkmark
the sum of the measures of $\angle abg$, $\angle fbe$, and $\angle dbc$ is always equal to $180^\circ$.
the measure of $\angle gbf$ is always equal to the measure of $\angle dbc$.
the sum of the measures of $\angle abg$, $\angle gbf$, $\angle fbe$, and $\angle dbc$ is always equal to $180^\circ$.

part b
solve for $x$ given that $m\angle abg = (2x + 3)^\circ$, $m\angle gbf = (5x + 9)^\circ$, and $m\angle fbe = 42^\circ$.

$x = \underline{\quad\quad\quad}$

Explanation:

Part A

To determine the truth value of each statement, we assume the angles are on a straight line (since the sum of angles on a straight line is \(180^\circ\)):

1. The sum of \( \angle ABG \), \( \angle GBF \), and \( \angle FBE \) is always \( 180^\circ \):

If \( ABE \) is a straight line, then \( \angle ABG + \angle GBF + \angle FBE = 180^\circ \) (linear pair/supplementary angles). Thus, this is True.

1. \( m\angle ABG = m\angle FBE \):

There is no information (e.g., vertical angles, bisectors) to guarantee \( \angle ABG \cong \angle FBE \). Thus, this is False (as marked).

1. The sum of \( \angle GBF \), \( \angle FBE \), and \( \angle DBC \) is always \( 180^\circ \):

\( \angle GBF + \angle FBE \) is part of \( \angle GBE \), but \( \angle DBC \) is unrelated to a straight line with them (no indication \( GBE \) and \( DBC \) form a straight line). Thus, this is False (as marked).

1. The sum of \( \angle ABG \), \( \angle FBE \), and \( \angle DBC \) is always \( 180^\circ \):

No guarantee these angles lie on a straight line. Thus, this is False.

1. \( m\angle GBF = m\angle DBC \):

No information (e.g., vertical angles, bisectors) to guarantee \( \angle GBF \cong \angle DBC \). Thus, this is False.

1. The sum of \( \angle ABG \), \( \angle GBF \), \( \angle FBE \), and \( \angle DBC \) is always \( 180^\circ \):

Four angles summing to \( 180^\circ \) is impossible (since three angles on a straight line sum to \( 180^\circ \); adding a fourth would exceed \( 180^\circ \) unless the fourth is \( 0^\circ \), which is not stated). Thus, this is False.

Part B

Step 1: Set up the equation

Since \( \angle ABG \), \( \angle GBF \), and \( \angle FBE \) lie on a straight line, their measures sum to \( 180^\circ \).
Given \( m\angle ABG = (2x + 3)^\circ \), \( m\angle GBF = (5x + 9)^\circ \), and \( m\angle FBE = 42^\circ \), we write:
$$(2x + 3) + (5x + 9) + 42 = 180$$

Step 2: Simplify and solve for \( x \)

Combine like terms:
$$2x + 5x + 3 + 9 + 42 = 180$$
$$7x + 54 = 180$$

Subtract 54 from both sides:
$$7x = 180 - 54$$
$$7x = 126$$

Divide by 7:
$$x = \frac{126}{7}$$
$$x = 18$$

Final Answers

Part A (True/False):

1. True
2. False (marked)
3. False (marked)
4. False
5. False
6. False

Part B:

Answer:

\( x = \boldsymbol{18} \)