The problem states that in the given figure $AB \parallel CD$, and you are required to find the value of $x$. The angles provided are labeled as:

• $\angle C = (3x + 13)^\circ$
• $\angle B = (2x - 23)^\circ$.

Since $AB \parallel CD$ and the lines are intersected by a transversal, we use the property of corresponding angles or alternate interior angles.

Step-by-Step Solution:

1. Identify the Relationship: Since $AB \parallel CD$, the angles $\angle C$ and $\angle B$ are alternate interior angles. Thus, they must be equal: $(3x + 13) = (2x - 23).$

2. Set up the Equation: Equating the angles: $3x + 13 = 2x - 23.$

3. Solve for $x$: Subtract $2x$ from both sides: $x + 13 = -23.$ Subtract $13$ from both sides: $x = -36.$

4. Verify the Result: Substitute $x = -36$ back into the expressions for $\angle C$ and $\angle B$: $\angle C = 3(-36) + 13 = -108 + 13 = -95^\circ,$ $\angle B = 2(-36) - 23 = -72 - 23 = -95^\circ.$ Both angles are equal, confirming the solution.

Final Answer:

$x = -36.$

Would you like a detailed explanation of any step? Let me know!

Related Questions:

1. What are the properties of parallel lines and transversals?
2. How are alternate interior angles formed in geometry?
3. Can $x$ have negative values in geometry, and what does it represent?
4. How would this problem change if the angles were supplementary instead of equal?
5. What other pairs of angles could we compare in parallel-line problems?

Tip:

In geometry, always verify the angle relationship (equal or supplementary) based on the given properties of parallel lines and transversals.