Let's solve the questions presented in the image.

Question 33:

We have two parallel lines cut by a transversal, and the expressions for two alternate interior angles are given:

• One angle is $11x + 1$ degrees.
• The other angle is $12x - 4$ degrees.

Since these are alternate interior angles and the lines are parallel, the angles are equal. Thus, we can set up the equation:

$11x + 1 = 12x - 4$

Solve for $x$:

$11x + 1 = 12x - 4$

Subtract $11x$ from both sides:

$1 = x - 4$

Add 4 to both sides:

$x = 5$

Now substitute $x = 5$ into either expression to find the angle. Using $11x + 1$:

$11(5) + 1 = 55 + 1 = 56^\circ$

So, the answer is C) 56°.

Question 34:

To make lines $u$ and $v$ parallel, the given angle and the unknown angle must be supplementary. This is because the angles form a pair of consecutive interior angles along the transversal.

The given angle is $99^\circ$, so the unknown angle $?$ is:

$180^\circ - 99^\circ = 81^\circ$

The answer is not listed in the provided options, which might suggest a mistake in the question setup.

Question 35:

Similarly, to make lines $u$ and $v$ parallel, the two consecutive interior angles (one of which is given as $100^\circ$) must add up to $180^\circ$. Therefore, the unknown angle $?$ is:

$180^\circ - 100^\circ = 80^\circ$

The answer is also not listed in the options, similar to the previous case.

Would you like further details on any of these steps or the concepts involved? Here are 5 related questions to deepen your understanding:

1. What are alternate interior angles, and why are they equal when lines are parallel?
2. How do consecutive interior angles relate to parallel lines and transversals?
3. Can you solve a similar problem where the angles are complementary rather than supplementary?
4. How does knowing one angle in a transversal diagram help you find other angles?
5. What methods can be used to prove that two lines are parallel?

Tip: When working with angles in parallel lines, always identify whether the angles are alternate, corresponding, or consecutive to decide the relationships!