Let's go through the image and solve the problems related to "Applying Angle Relationships." Here’s a breakdown of each section:

1. Parallel lines A and B are cut by transversal X

• Statement 1: "The marked angles are supplementary angles." (True)

  • This is correct because the angles on the same side of the transversal (interior angles) are supplementary, meaning they add up to 180°.

• Statement 2: "The marked angles are classified as same-side interior angles." (True)

  • This is also correct because the two angles formed between the parallel lines on the same side of the transversal are called same-side interior angles.

• Statement 3: "The value of x can be found using $4x - 23 = 2x + 5$." (False)

  • This equation is incorrect. The correct equation for supplementary angles would be: $(4x - 23) + (2x + 5) = 180°$ This simplifies to: $6x - 18 = 180 \quad \Rightarrow \quad 6x = 198 \quad \Rightarrow \quad x = 33$

2. Parallel lines A and B are cut by transversal X

• Statement 1: "The value of x = 24." (True)

  • We would verify this by solving the angle equations. \quad \Rightarrow \quad 9x = 219 \quad \Rightarrow \quad x = 24.33$$ This confirms the statement is approximately true, but since the question likely expects rounding, we can say $$x \approx 24$$.

• Statement 2: "The marked angles are congruent angles." (False)

  • This is false because the angles are supplementary, not congruent (congruent angles are equal in measure). The correct statement is: The marked angles are supplementary angles.

• Statement 3: "The marked angles are classified as corresponding angles." (True)

  • This is correct. These angles are corresponding angles because they occupy the same relative position at each intersection of the transversal with the parallel lines.

3. Parallel lines A and B are cut by transversal X

• Statement 1: "The value of x can be found using $12x - 18 = 8x + 10$." (True)

  • This is correct, as these are vertical angles, which are congruent (equal). Solving this equation: $12x - 18 = 8x + 10 \quad \Rightarrow \quad 4x = 28 \quad \Rightarrow \quad x = 7$

• Statement 2: "Each of the marked angles measures 56°." (False)

  • Solving for the angle using $x = 7$: $12(7) - 18 = 84 - 18 = 66°$ So, each angle measures 66°, not 56°. The correct statement is: Each of the marked angles measures 66°.

• Statement 3: "The marked angles are classified as vertical angles." (True)

  • This is true, as the angles are opposite each other at the intersection, which classifies them as vertical angles.

Would you like further explanation on any of the solutions? Here are some follow-up questions for practice:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. How would you solve for $x$ when angles are complementary?
3. Can two congruent angles be supplementary? Why or why not?
4. What’s the difference between corresponding angles and alternate exterior angles?
5. How do vertical angles help in finding unknown angle measures?

Tip: When working with angles, always check if the angles are supplementary (add to 180°) or congruent (equal) based on their position relative to parallel lines and transversals.