Let’s analyze both problems one by one.

Problem 11:

Here, we have two lines, $l$ and $m$, intersected by transversal lines. We can use the following geometric relationships:

1. Vertical Angles: Opposite angles are equal. For example, the angle labeled $49^\circ$ is equal to the angle labeled $(7x - 23)^\circ$. $49 = 7x - 23$ Solve for $x$: $49 + 23 = 7x \implies 72 = 7x \implies x = \frac{72}{7} \approx 10.29$

2. Corresponding Angles: Since $l$ and $m$ are parallel, the angles $(3x)^\circ$ and $(11y - 1)^\circ$ should be equal. $3x = 11y - 1$ Substitute $x$ from the previous calculation: $3 \left(\frac{72}{7}\right) = 11y - 1 \implies \frac{216}{7} = 11y - 1$ $\frac{216}{7} + 1 = 11y \implies \frac{223}{7} = 11y \implies y = \frac{223}{7 \cdot 11} \approx 2.9$

Problem 12:

In this triangle, we are given three angles. Since the sum of the angles in a triangle is $180^\circ$, we have:

$(7y - 20) + (5x - 38) + (3x - 4) = 90$

Step-by-step calculation:

1. Combine like terms: $7y - 20 + 5x - 38 + 3x - 4 = 180$ $7y + 8x - 62 = 180$ $7y + 8x = 242$

Now, let's summarize:

• For Problem 11, the values of $x$ and $y$ are approximately $x = 10.29$ and $y = 2.9$.
• For Problem 12, you can substitute these values into the corresponding relationships to verify or adjust the equations.

Would you like me to refine the calculations or provide further details on any steps?

Related Questions:

1. How are vertical angles formed, and why are they equal?
2. What are the different types of angle pairs formed by a transversal intersecting parallel lines?
3. What is the significance of the sum of angles in a triangle, and how does it help in solving geometric problems?
4. How do corresponding angles behave with parallel lines?
5. What is the general approach to solving equations involving multiple variables in geometric problems?

Tip:

When working with geometric figures and algebraic expressions together, it's crucial to identify angle relationships such as corresponding, alternate interior, and vertical angles, and then set up equations based on these relationships.