Let's analyze the problem step by step based on the image you've uploaded:

1) Top Right: Vertical Angles

• Given: Two intersecting lines forming vertical angles.
• The note states "vertical angles are" and implies that vertical angles are congruent. Therefore, the two opposite angles should be equal.
• One angle is given as 150°.
• Since vertical angles are congruent, the opposite angle will also be 150°.

2) Top Left: Vertical Angle Diagram with Algebra

• Two angles are shown:

  • One labeled $40^\circ$
  • The opposite angle (due to vertical angle property) will also be $40^\circ$.
  • Another unlabeled angle (likely complementary to 40°).

  If you're expected to find the remaining angles, the other pair of angles must add up to 180° because the two angles form a straight line (linear pair). Therefore: $180^\circ - 40^\circ = 140^\circ$ Hence, the other two vertical angles would both be 140°.

3) Bottom Right: Review, Solve for $x$

• Given:

  • $3x + 4$
  • $7x + 1$

  Since these angles form a straight line, they must add up to 180°. Set up the equation: $(3x + 4) + (7x + 1) = 180$ Simplify: $10x + 5 = 180$ Subtract 5 from both sides: $10x = 175$ Divide by 10: $x = 17.5$ So, the value of $x$ is 17.5.

4) Middle Problem: Line Segment (AB + BC = AC)

• Given:

  • $AB = 5x + 1$
  • $BC = 14$
  • $AC = 7x + 1$

  Since $AC = AB + BC$, the equation becomes: $(5x + 1) + 14 = 7x + 1$ Simplify the left-hand side: $5x + 15 = 7x + 1$ Subtract $5x$ from both sides: $15 = 2x + 1$ Subtract 1 from both sides: $14 = 2x$ Divide by 2: $x = 7$ So, the value of $x$ is 7.

5) Bottom Left: Angles in Triangle

• Given:

  • $m \angle AXC = 130^\circ$
  • $m \angle BXC = (2x)^\circ$

  Since the angles form a straight line, we know: $130^\circ + 2x = 180^\circ$ Solve for $x$: $2x = 180^\circ - 130^\circ = 50^\circ$ Divide by 2: $x = 25^\circ$ So, the value of $x$ is 25°.

Would you like further details or explanations on any of these?

Here are some related questions for deeper understanding:

1. What are some properties of vertical angles?
2. How can linear pairs help solve for unknown angles?
3. How do you solve an equation involving segments and algebra?
4. What is the relationship between angles on a straight line?
5. How can geometry help in solving algebraic expressions involving angles?

Tip: Always look for congruent or supplementary angle relationships to help with unknowns!