To solve this problem, we have to find the value of \( x \) using the angle addition postulate, and then we will determine the measure of angle \( BAC \).
Part A: Write and solve an equation for \( x \)
We are given:
- \( m\angle BAD = 129^\circ \)
- \( m\angle BAC = (2x - 1)^\circ \)
- \( m\angle CAD = (3x + 5)^\circ \)
Since angles \( BAC \) and \( CAD \) are adjacent angles that share a common ray \( AC \), we can express the entire angle \( BAD \) as the sum of angles \( BAC \) and \( CAD \):
\[ m\angle BAD = m\angle BAC + m\angle CAD \]
Substituting the expressions we have:
\[ 129^\circ = (2x - 1)^\circ + (3x + 5)^\circ \]
Now, combine the expressions on the right side:
\[ 129 = (2x - 1) + (3x + 5) \]
Simplifying the right side:
\[ 129 = 2x - 1 + 3x + 5 \] \[ 129 = 5x + 4 \]
Now, isolate \( x \):
\[ 129 - 4 = 5x \] \[ 125 = 5x \] \[ x = \frac{125}{5} = 25 \]
Thus, we have:
\[ x = 25 \]
Part B: Find \( m\angle BAC \)
Now that we've found \( x \), we can substitute it back into the formula for \( m\angle BAC \):
\[ m\angle BAC = 2x - 1 \]
Substituting \( x = 25 \):
\[ m\angle BAC = 2(25) - 1 \] \[ m\angle BAC = 50 - 1 \] \[ m\angle BAC = 49^\circ \]
Therefore, the measure of angle \( BAC \) is:
\[ m\angle BAC = 49^\circ \]
Summary of Results
- Part A: \( x = 25 \)
- Part B: \( m\angle BAC = 49^\circ \)