Question

current skill $ what is the value of x?

Explanation:

Step1: Identify Alternate Interior Angles

The angles \(4x + 3\) and \(3x + 2\) are alternate interior angles, so they are equal.

$$4x + 3 = 3x + 2$$

Step2: Solve for \(x\)

Subtract \(3x\) from both sides:

$$4x - 3x + 3 = 3x - 3x + 2$$

$$x + 3 = 2$$

Subtract 3 from both sides:

$$x + 3 - 3 = 2 - 3$$

$$x = -1$$

Wait, that can't be right. Wait, maybe they are corresponding angles? Wait, no, maybe I misidentified. Wait, actually, the two horizontal lines are parallel? Wait, the transversal cuts two parallel lines, so corresponding angles. Wait, no, maybe the angles \(4x + 3\) and \(3x + 2\) are same - side? No, wait, maybe I made a mistake. Wait, actually, the two angles \(4x + 3\) and \(3x + 2\) are consecutive interior angles? No, wait, let's look again. Wait, the two horizontal lines: the upper horizontal line and the lower horizontal line. The transversal is the slant line. So \(4x + 3\) and \(3x + 2\) are actually same - side? No, wait, maybe they are supplementary? Wait, no, if the lines are parallel, then alternate interior angles are equal. Wait, maybe the lines are parallel, so \(4x + 3\) and \(3x + 2\) are equal? Wait, but when I solved, I got \(x=-1\), which would make \(4x + 3=-4 + 3=-1\) and \(3x + 2=-3 + 2=-1\), but angles can't be negative. So I must have misidentified the angles. Wait, maybe \(4x + 3\) and \(3x + 2\) are supplementary? Wait, no, maybe the angles \(4x + 3\) and \(3x + 2\) are actually vertical angles? No, vertical angles are opposite. Wait, maybe the two horizontal lines are parallel, so the sum of \(4x + 3\) and \(3x + 2\) is 180? Wait, no, consecutive interior angles. Wait, let's check the diagram again. The upper horizontal line: angles 1,2,3,4x + 3. The lower horizontal line: angles 5,3x + 2,7,8. The transversal is the slant line. So angle 3 and angle 5 are corresponding angles? Wait, no, angle 4x + 3 and angle 3x + 2: maybe they are same - side interior angles, so their sum is 180. Wait, let's try that. So \(4x + 3+3x + 2 = 180\)

$$7x+5 = 180$$

$$7x=180 - 5$$

$$7x = 175$$

$$x=\frac{175}{7}=25$$

Ah, that makes sense. I must have misidentified the angle relationship. So the correct relationship is that \(4x + 3\) and \(3x + 2\) are consecutive interior angles (same - side interior angles) because the two horizontal lines are parallel, so consecutive interior angles are supplementary.

So Step 1: Identify the angle relationship. Since the two horizontal lines are parallel (implied by the diagram, as it's a typical parallel lines and transversal problem) and the slant line is a transversal, \(4x + 3\) and \(3x + 2\) are consecutive interior angles, so they are supplementary. So:

$$4x + 3+3x + 2=180$$

Step 2: Combine like terms.

$$7x + 5 = 180$$

Step 3: Subtract 5 from both sides.

$$7x+5 - 5=180 - 5$$

$$7x = 175$$

Step 4: Divide both sides by 7.

$$x=\frac{175}{7}=25$$

Answer:

\(25\)