Question

find the value of x.
5)
6)
ecuación:
ecuación:
x =
x =

Explanation:

Problem 5

Step 1: Identify the relationship

The two angles are same - side interior angles or supplementary? Wait, actually, when two parallel lines are cut by a transversal, same - side interior angles are supplementary, but also, the angle \(105^{\circ}\) and \((4x + 1)^{\circ}\) are same - side interior angles? Wait, no, actually, the angle \(105^{\circ}\) and the angle adjacent to \((4x + 1)^{\circ}\) (vertical angles or corresponding) - Wait, actually, if we consider that the two angles \(105^{\circ}\) and \((4x + 1)^{\circ}\) are same - side interior angles, they should be supplementary? Wait, no, let's think again. If the lines are parallel, then same - side interior angles are supplementary. But \(105+(4x + 1)=180\)? Wait, no, maybe they are alternate interior angles? Wait, no, the angle \(105^{\circ}\) and \((4x + 1)^{\circ}\) - Wait, maybe the angle \(105^{\circ}\) and \((4x + 1)^{\circ}\) are supplementary. Let's check:

If two parallel lines are cut by a transversal, same - side interior angles are supplementary. So \(105+(4x + 1)=180\)

Step 2: Solve the equation

\(105 + 4x+1=180\)

Combine like terms: \(106+4x = 180\)

Subtract 106 from both sides: \(4x=180 - 106=74\)

Divide both sides by 4: \(x=\frac{74}{4}=18.5\)? Wait, no, wait, maybe I made a mistake. Wait, maybe the angles are equal? Wait, no, \(105^{\circ}\) and \((4x + 1)^{\circ}\) - Wait, maybe they are alternate exterior angles? No, let's re - examine the diagram.

Wait, actually, the angle \(105^{\circ}\) and \((4x + 1)^{\circ}\) are same - side interior angles, so they should add up to \(180^{\circ}\). So:

\(105+(4x + 1)=180\)

\(4x+106 = 180\)

\(4x=180 - 106=74\)

\(x=\frac{74}{4}=18.5\)? Wait, that seems odd. Wait, maybe the angle \(105^{\circ}\) and \((4x + 1)^{\circ}\) are supplementary because they are same - side interior angles.

Wait, let's check again. If the lines are parallel, same - side interior angles sum to \(180^{\circ}\). So the equation is \(4x + 1+105 = 180\)

Step 3: Solve for x

\(4x+106 = 180\)

\(4x=180 - 106 = 74\)

\(x=\frac{74}{4}=18.5\)

Problem 6

Step 1: Identify the relationship

The two angles \(4x^{\circ}\) and \(84^{\circ}\) are same - side interior angles? Wait, no, if the lines are parallel, and the transversal cuts them, then \(4x\) and \(84\) are same - side interior angles? Wait, no, maybe they are supplementary? Wait, no, \(4x+84 = 180\)? Wait, no, maybe \(4x = 84\)? Wait, no, if they are corresponding angles? Wait, no, let's think. If the lines are parallel, and the transversal is such that \(4x\) and \(84\) are same - side interior angles, then \(4x + 84=180\). Wait, no, maybe \(4x=84\)? No, that would be if they are alternate interior angles. Wait, maybe the diagram shows that \(4x\) and \(84\) are same - side interior angles, so they are supplementary.

Step 2: Set up the equation

\(4x+84 = 180\)

Step 3: Solve for x

Subtract 84 from both sides: \(4x=180 - 84 = 96\)

Divide both sides by 4: \(x = 24\)

Final Answers

Problem 5

Step 1: Determine the equation

Since the lines are parallel and cut by a transversal, the angles \((4x + 1)^{\circ}\) and \(105^{\circ}\) are same - side interior angles, so they are supplementary. Thus, the equation is \(4x + 1+105=180\).

Step 2: Solve the equation

\(4x+106 = 180\)

\(4x=180 - 106=74\)

\(x=\frac{74}{4}=18.5\)

Answer:

Equation: \(4x + 1+105 = 180\)

\(x = 18.5\)

Problem 6