Let's go through the two problems from the image.

Problem 27:

We need to find the value of $x$ for which the two lines $\ell$ and $m$ are parallel. To do this, we can use the fact that alternate interior angles or corresponding angles are equal when lines are parallel.

In the diagram for Problem 27:

• There is an angle of $27x^\circ$ and an angle of $19x^\circ$ on opposite sides of the transversal. These angles are alternate interior angles, so they should be equal when the lines are parallel. Therefore, we can set up the equation:

$27x = 19x + 17^\circ$

Now solve for $x$:

$27x - 19x = 17$

$8x = 17$

$x = \frac{17}{8} = 2.125$

So, the value of $x$ is approximately $2.125$.

Problem 28:

For this problem, again we need to find $x$ such that the lines $\ell$ and $m$ are parallel.

In the diagram:

• The angle $5x^\circ$ and the angle $5x + 40^\circ$ are corresponding angles, so they must be equal when the lines are parallel. Set up the equation:

$5x + 40 = 2x$

Now solve for $x$:

$5x + 40 = 2x$

$5x - 2x = -40$

$3x = -40$

$x = \frac{-40}{3} = -\frac{40}{3} \approx -13.33$

So, the value of $x$ is approximately $-13.33$.

Would you like any further clarification or detailed explanation on these problems?

Here are 5 related questions:

1. What are alternate interior angles, and how do they relate to parallel lines?
2. What are corresponding angles, and why are they equal when lines are parallel?
3. How can we use angle relationships to prove that two lines are parallel?
4. What would happen if the given lines were not parallel in the problems above?
5. What other methods can be used to determine if two lines are parallel?

Tip: In geometry, recognizing angle relationships (like alternate interior or corresponding angles) is key to proving lines are parallel or calculating unknown values.