In the image, we have two angles formed by intersecting lines, with the measures of the angles given in terms of $x$.

The angles are marked as:

• $(4x + 1)^\circ$ (top-left)
• $(2x - 19)^\circ$ (bottom-left)

These angles are vertical angles, which means they are congruent (equal to each other). Therefore, we can set their expressions equal and solve for $x$:

$4x + 1 = 2x - 19$

Step-by-step solution:

1. Subtract $2x$ from both sides:

$4x - 2x + 1 = -19$ $2x + 1 = -19$

2. Subtract 1 from both sides:

$2x = -19 - 1$ $2x = -20$

3. Divide both sides by 2:

$x = \frac{-20}{2}$ $x = -10$

So, the value of $x$ is $-10$.

Would you like further explanation on this, or have any additional questions?

Here are 5 related questions for practice:

1. How do you solve for $x$ when two supplementary angles are given?
2. How can you use the properties of corresponding angles to solve equations?
3. What is the difference between vertical and adjacent angles?
4. How do you find the value of $x$ when angles are complementary?
5. How do you solve for an unknown angle in an isosceles triangle?

Tip: Always look for angle relationships like vertical, corresponding, or supplementary when solving angle problems involving intersecting lines!