Question

1. \\( x = \\) \\( (5x)\degree \\) \\( (16x - 9)\degree \\) \\( (8x - 23)\degree \\) \\( (6x + 15)\degree \\)

Explanation:

Problem 5 (Assuming the angles are equal, maybe alternate interior angles)

Step1: Set the angles equal

Since the lines are parallel (implied by the arrows) and the angles are alternate interior angles, we set \(5x = 16x - 9\).

Step2: Solve for \(x\)

Subtract \(5x\) from both sides: \(0 = 11x - 9\).
Add 9 to both sides: \(11x = 9\).
Divide by 11: \(x=\frac{9}{11}\)? Wait, maybe I misread. Wait, maybe the other problem (7) first. Wait, let's check problem 7.

Problem 7 (Vertical angles or alternate interior? Wait, the angles \((8x - 23)^\circ\) and \((6x + 15)^\circ\) look like vertical angles or alternate interior if lines are parallel. Let's assume they are equal (vertical angles or alternate interior for parallel lines).

Step1: Set the angles equal

\(8x - 23 = 6x + 15\)

Step2: Solve for \(x\)

Subtract \(6x\) from both sides: \(2x - 23 = 15\).
Add 23 to both sides: \(2x = 38\).
Divide by 2: \(x = 19\).

Wait, maybe problem 5: Let's re-examine. If the angles are same - side interior angles, they would be supplementary. Wait, maybe I made a mistake. Let's check the diagram again. The first problem (5) has angles \(5x\) and \(16x - 9\). If they are same - side interior angles, \(5x+(16x - 9)=180\).

Step1: Set up the equation

\(5x + 16x - 9 = 180\)

Step2: Combine like terms

\(21x - 9 = 180\)

Step3: Solve for \(x\)

Add 9 to both sides: \(21x = 189\)
Divide by 21: \(x = 9\). Ah, that makes more sense. So probably same - side interior angles (supplementary).

For Problem 5 (Correcting):

Step1: Supplementary angles (same - side interior)

\(5x+(16x - 9)=180\)

Step2: Simplify

\(21x - 9 = 180\)

Step3: Add 9 to both sides

\(21x = 189\)

Step4: Divide by 21

\(x = 9\)

For Problem 7:

Step1: Equal angles (vertical or alternate interior)

\(8x - 23 = 6x + 15\)

Step2: Subtract \(6x\)

\(2x - 23 = 15\)

Step3: Add 23

\(2x = 38\)

Step4: Divide by 2

\(x = 19\)

Answer:

s:
Problem 5: \(x = 9\)
Problem 7: \(x = 19\)