Let me help you with the math problems presented in the image. The picture shows multiple problems involving angles with transversal lines and parallel lines.

Here’s a breakdown of the problems:

Problem 3:

Find the sizes of angles $a$, $b$, $c$, and $d$ based on a diagram that shows intersecting lines.

The diagram shows that $a = 140^\circ$, and the remaining angles are calculated using the fact that:

• Vertically opposite angles are equal.
• Adjacent angles on a straight line sum up to $180^\circ$.

Hence:

• $b = 180^\circ - a = 180^\circ - 140^\circ = 40^\circ$,
• $c = b = 40^\circ$ (as they are vertically opposite),
• $d = 180^\circ - b = 180^\circ - 40^\circ = 140^\circ$ (as it is supplementary to angle $b$).

Problem 4:

Find the sizes of all the angles in the given figure.

The figure shows intersecting lines with one given angle $\angle FKC = 130^\circ$. Based on the rules of intersecting lines:

• Vertically opposite angles are equal, so $\angle FKE = 130^\circ$,
• Adjacent angles on a straight line sum up to $180^\circ$, so $\angle KBE = 50^\circ$ and $\angle KAC = 50^\circ$,
• $\angle KFD = \angle FKE = 130^\circ$.

Problem 5:

Find the sizes of all angles in the diagram involving two transversals crossing two sets of parallel lines.

Here, $\angle AMP = 95^\circ$. Using the properties of parallel lines cut by a transversal:

• Corresponding angles are equal,
• Alternate interior angles are equal,
• Consecutive interior angles sum up to $180^\circ$.

Let me know if you'd like a more detailed explanation for any of these problems, and here are some additional questions that you might find useful for further understanding:

1. Why are vertically opposite angles always equal?
2. How do corresponding angles behave when a transversal crosses parallel lines?
3. What is the sum of the interior angles on the same side of a transversal when it intersects parallel lines?
4. How are alternate interior angles identified when two parallel lines are crossed by a transversal?
5. In which scenarios do you use the rule that adjacent angles on a straight line sum to 180°?

Tip: Always check if two lines are parallel, as this can simplify angle calculations when dealing with transversals.