In the given image, we have two parallel lines $m$ and $n$, cut by a transversal. The angles formed are:

• $(3x - 18)^\circ$ at the upper left intersection,
• $y^\circ$ at the upper right intersection,
• $(2x + 8)^\circ$ at the lower left intersection.

Step 1: Identify angle relationships

Since lines $m$ and $n$ are parallel, the corresponding angles are equal. Thus, the angle $(3x - 18)^\circ$ formed at the top left intersection corresponds to the angle $(2x + 8)^\circ$ at the bottom left intersection. These angles are congruent because they are corresponding angles formed by a transversal with parallel lines.

So, we can set up the equation: $3x - 18 = 2x + 8$

Step 2: Solve for $x$

We now solve the equation: $3x - 18 = 2x + 8$ Subtract $2x$ from both sides: $x - 18 = 8$ Add 18 to both sides: $x = 26$

Step 3: Calculate the angles

Substitute $x = 26$ into the expressions for the angles:

• $(3x - 18)^\circ = 3(26) - 18 = 78 - 18 = 60^\circ$,
• $(2x + 8)^\circ = 2(26) + 8 = 52 + 8 = 60^\circ$.

Thus, the angles are both $60^\circ$, which confirms the correctness of the solution since corresponding angles are equal.

Step 4: Calculate $y$

The angle $y$ is vertically opposite to $(3x - 18)^\circ$. Since vertically opposite angles are equal: $y = 60^\circ$

Final Answer:

• $x = 26$,
• The angles $(3x - 18)^\circ = 60^\circ$,
• $(2x + 8)^\circ = 60^\circ$,
• $y = 60^\circ$.

Would you like more details or have any questions?

Here are 5 related questions:

1. What are corresponding angles in parallel line transversals?
2. Can vertically opposite angles ever be unequal?
3. How do we identify alternate interior angles in parallel lines?
4. What happens to the angle values if the lines are not parallel?
5. How can we solve for unknown angles if only one angle is given?

Tip: Always look for relationships like corresponding or alternate interior angles when dealing with parallel lines and a transversal.