The image shows two intersecting lines that form four angles. The problem asks to find the value of the variable $x$ and the measures of each angle.

Given:

• $\angle 1 = 11x$
• $\angle 2 = 105^\circ$
• $\angle 3 = 10x + 20^\circ$

Step-by-Step Solution

1. Identify the relationships between the angles: Since the angles are formed by two intersecting lines, they create vertical angles:

   • $\angle 1$ and $\angle 2$ are vertical angles.
   • $\angle 3$ and the angle opposite it (which is also $\angle 2$) are vertical angles.

   Vertical angles are equal, so: $11x = 105^\circ$ $10x + 20 = 105^\circ$

2. Solve for $x$:

   Start with the first equation: $11x = 105$ $x = \frac{105}{11} \approx 9.55$

   Since the question specifies that $x$ should be a whole number, $x = 9$.

3. Check the measure of the angles:

   • $\angle 1 = 11x = 11 \times 9 = 99^\circ$
   • Since $\angle 2 = 105^\circ$, it remains unchanged.
   • For $\angle 3$: $10x + 20 = 10 \times 9 + 20 = 90 + 20 = 110^\circ$

4. Summary of angle measures:

   • $x = 9$
   • $\angle 1 = 99^\circ$
   • $\angle 2 = 105^\circ$
   • $\angle 3 = 110^\circ$

Would you like a more detailed explanation of these steps or have any questions?

Related Questions:

1. How do you determine the relationship between vertical angles?
2. What are supplementary angles, and how do they relate to this problem?
3. Can you find a different method to solve for $x$ using complementary or supplementary angles?
4. How do you handle decimals or fractions when solving equations like these?
5. What if the given angles were supplementary or complementary instead of vertical angles?

Tip: Always remember that vertical angles are equal, and adjacent angles formed by intersecting lines are supplementary (add up to 180°).