The image shows six geometry problems where you are tasked with finding the values of unknown variables (like $x$, $y$, and $m$) and identifying the theorems used. Here's how to approach each one:

1. Problem 1

Given:

• One angle is $61^\circ$, and the other angle is labeled as $y$.

Observation: These angles are on a straight line, forming a linear pair.

Theorem: Linear Pair Theorem states that the sum of angles on a straight line is $180^\circ$.

Solution: $61^\circ + y = 180^\circ$ $y = 180^\circ - 61^\circ = 119^\circ$

2. Problem 2

Given:

• Two angles: $8x^\circ$ and $96^\circ$.

Observation: These angles are vertical angles, which are congruent.

Theorem: Vertical Angle Theorem.

Solution: $8x = 96$ $x = \frac{96}{8} = 12$

3. Problem 3

Given:

• Two expressions for angles: $8x + 2^\circ$ and $66(2y - 3)^\circ$.

Observation: The angles are corresponding angles (parallel lines are indicated), which means they are equal.

Theorem: Corresponding Angles Postulate.

Solution: Equating the two expressions: $8x + 2 = 66(2y - 3)$ First, simplify the right side: $8x + 2 = 132y - 198$ Now you would solve for $x$ and $y$ step by step.

4. Problem 4

Given:

• Two angles: $x^\circ$ and $97^\circ$.

Observation: These angles are alternate interior angles, which are congruent.

Theorem: Alternate Interior Angles Theorem.

Solution: $x = 97^\circ$

5. Problem 5

Given:

• Two expressions: $4x + 24^\circ$ and (8x