In the image provided, you have two angles that are given along the line:

1. $\angle ABC = (5x - 22)^\circ$
2. $\angle BCD = (8x + 46)^\circ$

Since $\angle ABC$ and $\angle BCD$ form a linear pair (they are on a straight line), their sum must equal $180^\circ$. This is because adjacent angles on a straight line are supplementary.

Let's solve for $x$:

$(5x - 22) + (8x + 46) = 180$

Simplifying this equation:

$5x - 22 + 8x + 46 = 180$

$13x + 24 = 180$

Subtract 24 from both sides:

$13x = 156$

Now, divide by 13:

$x = 12$

Now that we know $x = 12$, we can find the measure of the angles:

$\angle ABC = 5(12) - 22 = 60 - 22 = 38^\circ$ $\angle BCD = 8(12) + 46 = 96 + 46 = 142^\circ$

Thus, $\angle ABC = 38^\circ$ and $\angle BCD = 142^\circ$.

Would you like further details or explanations?

Here are 5 related questions to consider:

1. What is the significance of supplementary angles?
2. How can you verify if two angles form a linear pair?
3. What happens to the sum of angles when three or more angles are adjacent on a straight line?
4. How can we use the concept of vertical angles in similar problems?
5. Can angles with different expressions (in terms of $x$) ever form complementary angles?

Tip: Always check your results by substituting the values back into the original expressions to ensure they satisfy the given conditions.