Two angles, $\angle 1$ and $\angle 2$, are supplementary. The measure of $\angle 1$ is $(4x - 23)^\circ$ and the measure of $\angle 2$ is $(2x + 11)^\circ$. We need to find the measure of $\angle 1$.

1. Problem Description

Two angles, $\angle 1$ and $\angle 2$, are supplementary. The measure of $\angle 1$ is $(4x - 23)^\circ$ and the measure of $\angle 2$ is $(2x + 11)^\circ$. We need to find the measure of $\angle 1$.

2. Solution Steps

Since $\angle 1$ and $\angle 2$ are supplementary angles, their measures add up to $180^\circ$. Therefore, we can write the equation:

$$m\angle 1 + m\angle 2 = 180^\circ$$

Substitute the given expressions for the measures of the angles:

$$(4x - 23) + (2x + 11) = 180$$

Combine like terms:

$$6x - 12 = 180$$

Add 12 to both sides of the equation:

$$6x = 192$$

Divide both sides by 6:

$$x = \frac{192}{6} = 32$$

Now that we have found the value of $x$, we can find the measure of $\angle 1$:

$$m\angle 1 = 4x - 23$$

Substitute $x = 32$:

$$m\angle 1 = 4(32) - 23$$

$$m\angle 1 = 128 - 23$$

$$m\angle 1 = 105$$

Therefore, the measure of $\angle 1$ is $105^\circ$.

3. Final Answer

$105^\circ$