Let's solve for the values of $x$, $y$, and $z$ using the properties of angles formed by intersecting lines.

Step 1: Identify angle relationships

1. The two intersecting lines form vertically opposite angles, which are always equal. Hence: $3x - 10^\circ = y \quad \text{(since they are vertically opposite angles)}$ Similarly, for the angles on the other side of the intersection: $2x + 15^\circ = z \quad \text{(since they are vertically opposite angles)}$

2. The adjacent angles along a straight line must add up to $180^\circ$ because they are supplementary. Therefore: $(3x - 10^\circ) + (2x + 15^\circ) = 180^\circ$ Simplifying this equation: $5x + 5^\circ = 180^\circ$ Subtract $5^\circ$ from both sides: $5x = 175^\circ$ Divide by 5: $x = 35^\circ$

Step 2: Find $y$ and $z$

Now that we know $x = 35^\circ$, we can substitute this value back into the expressions for $y$ and $z$.

1. For $y$: $y = 3x - 10^\circ = 3(35^\circ) - 10^\circ = 105^\circ - 10^\circ = 95^\circ$

2. For $z$: $z = 2x + 15^\circ = 2(35^\circ) + 15^\circ = 70^\circ + 15^\circ = 85^\circ$

Final Answer:

• $x = 35^\circ$
• $y = 95^\circ$
• $z = 85^\circ$

Reasons for Steps:

• Vertically opposite angles: These are equal, so we used this to relate $y$ and $z$ to the given angles.
• Supplementary angles: Angles on a straight line add up to $180^\circ$, which allowed us to form an equation to solve for $x$.

Let me know if you want further clarification or details on any part of the solution.

Related Questions:

1. What are complementary angles, and how are they different from supplementary angles?
2. Can two obtuse angles ever be supplementary? Why or why not?
3. How do we define alternate interior angles, and when are they equal?
4. What is the sum of angles at a point, and how is it applied in geometry?
5. If a transversal intersects two parallel lines, how can we use angle properties to find unknown angles?

Tip:

Always look for vertical angles or supplementary angles when working with intersecting lines, as they often provide the key to solving for unknowns.