This is a geometry problem involving angles, and we need to find the values of $x$ and $y$. Here's the breakdown of the solution:

Observations

1. Straight Line Rule: The sum of angles on a straight line is $180^\circ$.
2. Triangle Angle Sum: The sum of the angles in a triangle is $180^\circ$.

From the given diagram:

• The angles on the straight line include $81^\circ$ and $6x - 11^\circ$, so: $81 + (6x - 11) = 180$

• The three angles in the left triangle are: $(4y - 18)^\circ, (y + 14)^\circ, \text{and } 81^\circ.$ Thus, the sum of these angles is $180^\circ$: $(4y - 18) + (y + 14) + 81 = 180$

• In the right triangle: $(x + 5)^\circ, (6x - 11)^\circ, \text{and the exterior angle } 81^\circ.$

We will solve for $x$ and $y$.

Step 1: Solve for $x$

Using the straight line equation: $81 + (6x - 11) = 180$ Simplify: $6x + 70 = 180$ $6x = 110 \quad \Rightarrow \quad x = \frac{110}{6} = 18.33^\circ \ (\text{rounded to two decimal places}).$

Step 2: Solve for $y$

Using the triangle equation in the left triangle: $(4y - 18) + (y + 14) + 81 = 180$ Simplify: $4y - 18 + y + 14 + 81 = 180$ $5y + 77 = 180$ $5y = 103 \quad \Rightarrow \quad y = \frac{103}{5} = 20.6^\circ.$

Final Values:

• $x = 18.33^\circ$
• $y = 20.6^\circ$

Let me know if you'd like a detailed verification or further questions! Here are related questions:

Related Questions:

1. How would the solution change if the straight-line angle was altered?
2. Can you generalize a formula for calculating angles in such diagrams?
3. How would this problem differ with exterior angle sums of a polygon?
4. Why is it crucial to check triangle and straight-line rules in such problems?
5. What are common errors to avoid in angle calculations?

Tip:

Always double-check your equations, especially when combining multiple conditions like straight-line angles and triangle sums!