https://file.aiquickdraw.com/mathbot/file/17265279559669wjzws10.jpg

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.

In the diagram, two angles are given along a straight line:
- Angle ABC = (5x - 22)°
- Angle BCD = (8x + 46)°
Find the values of x, Angle ABC, and Angle BCD.

Learn how to solve for unknown angles using linear pairs and algebraic expressions. This example problem walks you through solving (5x - 22)° and (8x + 46)° on a straight line, demonstrating how the sum of angles equals 180°. Suitable for students in grades 9-10.

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