The problem involves a figure with two intersecting lines labeled as $a$ and $b$. The angle expressions are given at the intersection:

• Angle $(5x - 7)^\circ$
• Angle $(3x + 17)^\circ$
• Angle $(4y + 3)^\circ$

Analysis

These angles are formed by intersecting lines, making the opposite (vertical) angles equal. Using this property, we can set up equations:

1. Vertical Angles: $5x - 7 = 4y + 3$

2. Adjacent Angles (on a straight line sum to $180^\circ$): $(5x - 7) + (3x + 17) = 180$

Solving the Equations

1. From the adjacent angles equation: $(5x - 7) + (3x + 17) = 180$ Simplifying: $8x + 10 = 180 \implies 8x = 170 \implies x = 21.25$

2. Substitute $x = 21.25$ into the vertical angles equation: $5(21.25) - 7 = 4y + 3$ Simplifying: $106.25 - 7 = 4y + 3 \implies 99.25 = 4y + 3 \implies 4y = 96.25 \implies y = 24.06$

Conclusion

The values of $x$ and $y$ do not exactly match any of the options provided. However, the closest match is:

• $x = 22.85, y = 22.86$.

If you have any questions or need further clarification, let me know!

Relative Questions:

1. How do vertical angles relate to each other in intersecting lines?
2. How can we verify that the sum of angles on a straight line equals $180^\circ$?
3. What would be a different approach to solving for $x$ and $y$ without using vertical angles?
4. What are the properties of angles formed by parallel and transversal lines?
5. How would the problem change if there were no equal angles?

Tip:

Always remember to check for angle relationships like vertical angles, corresponding angles, and linear pairs when dealing with intersecting or parallel lines.