Let's go through the problems on the sheet you uploaded. I will solve them step-by-step:

Find the measure of angle $b$:

Problem 13:

• There is a straight line with angles 65° and $b$ formed by intersecting lines.

• The sum of the angles on a straight line is 180°.

  So, $b = 180^\circ - 65^\circ = 115^\circ$.

Problem 14:

• This is a Z-shaped alternate interior angle problem. The angles on opposite sides of the transversal line are equal.
• So, $b = 25^\circ$ (alternate interior angles are congruent).

Problem 15:

• This shows two vertical angles, one of which is given as 75°. Vertical angles are equal.

  Hence, $b = 75^\circ$.

Problem 16:

• This is another straight-line scenario where $b$ and the 133° angle are supplementary.

  So, $b = 180^\circ - 133^\circ = 47^\circ$.

Find the value of $x$:

Problem 17:

• There are vertically opposite angles, $41^\circ$ and $(2x + 1)^\circ$.

• Since vertically opposite angles are equal:

  $2x + 1 = 41$ $2x = 40$ $x = 20$

Problem 18:

• We have a linear pair of angles that form a straight line, and their sum must be 180°:

  $(5x + 3) + (4x + 6) = 180$ Simplifying: $9x + 9 = 180$ $9x = 171$ $x = 19$

Problem 19:

• This triangle has one angle as $80^\circ$ and another as $5x^\circ$.

• Since the sum of angles in a triangle is always 180°:

  $80 + 5x + 90 = 180$ Simplifying: $5x + 80 = 180$ $5x = 100$ $x = 20$

Problem 20:

• Another triangle problem. The angle measures are $(2x + 18)^\circ$, $(3x + 2)^\circ$, and the right angle $90^\circ$.

• Again, using the sum of angles in a triangle:

  $(2x + 18) + (3x + 2) + 90 = 180$ Simplifying: $5x + 20 + 90 = 180$ $5x + 110 = 180$ $5x = 70$ $x = 14$

Would you like more details or clarifications for any specific problem?

Here are 5 related questions to expand your understanding:

1. What are supplementary angles and how are they used in geometry?
2. How can we identify vertical angles in more complex diagrams?
3. How do alternate interior angles help solve for unknown angles?
4. Why do the sum of angles in a triangle always add up to 180°?
5. How does knowing the properties of different angle pairs simplify problem solving?

Tip: Always look for relationships like vertical angles or straight-line angles to solve for unknowns easily.