Let's examine both diagrams and solve for the measure of $\angle ABC$ in each case.

Left Diagram

1. In the left diagram, two parallel lines are crossed by two transversals forming multiple angles around points $A$, $B$, and $C$.

2. We know the following angle measures:

   • $\angle A = 140^\circ$
   • $\angle C = 62^\circ$

3. Using the properties of parallel lines and the fact that angles on a straight line sum up to $180^\circ$:

   • $\angle ABC$ is formed by the intersection of the transversal with the parallel lines. It is an alternate interior angle to $\angle A$.
   • Therefore, $\angle ABC = 140^\circ$.

Right Diagram

1. In the right diagram, we have:

   • $\angle A = 30^\circ$
   • $\angle C = 50^\circ$

2. To find $\angle ABC$:

   • Since these are consecutive interior angles (or co-interior angles) formed by a transversal intersecting parallel lines, they should sum up to $180^\circ$.
   • Thus, $\angle ABC = 180^\circ - (30^\circ + 50^\circ)$.
   • Calculating this, we find: $\angle ABC = 180^\circ - 80^\circ = 100^\circ.$

Summary of Results

• For the left diagram: $\angle ABC = 140^\circ$.
• For the right diagram: $\angle ABC = 100^\circ$.

Would you like further clarification on any of the steps?

Related Questions:

1. What is an alternate interior angle, and why is it used here?
2. How do co-interior angles help in calculating angles in parallel line problems?
3. Why do angles on a straight line sum to $180^\circ$?
4. What other angle properties can be derived from parallel lines?
5. Can this method be applied to find unknown angles in polygons?

Tip: In parallel line problems, identifying angle pairs (alternate interior, corresponding, or co-interior) helps in determining unknown angles.