In this image, we have two parallel lines $p \parallel q$ and two transversal lines $l$ and $m$, creating multiple angles. The question asks to find the measures of angles $a$, $b$, $c$, and $d$, with some given angles already provided: $\angle a = 110^\circ$ and $\angle d = 115^\circ$.

Step-by-Step Solution:

1. Angle $a$: It is already given that $\angle a = 110^\circ$.

2. Angle $b$: Since lines $p \parallel q$, and line $l$ is a transversal, $\angle a$ and $\angle b$ are corresponding angles. Corresponding angles are equal when a transversal intersects parallel lines. $\therefore \angle b = \angle a = 110^\circ$

3. Angle $c$: The given $\angle d = 115^\circ$ and lines $q \parallel p$, with line $m$ as the transversal, imply that $\angle c$ and $\angle d$ are alternate interior angles. Alternate interior angles are equal when the lines are parallel. $\therefore \angle c = \angle d = 115^\circ$

4. Angle $d$: It is already provided that $\angle d = 115^\circ$.

Summary of the Angle Measures:

• $\angle a = 110^\circ$
• $\angle b = 110^\circ$
• $\angle c = 115^\circ$
• $\angle d = 115^\circ$

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Here are 5 follow-up questions related to this concept:

1. What is the difference between corresponding angles and alternate interior angles?
2. How can you use the concept of vertically opposite angles in this diagram?
3. What are co-interior angles, and are there any in this diagram?
4. What happens to the angles if the transversal lines are no longer perpendicular to the parallel lines?
5. How would the angles change if the parallel lines were non-parallel?

Tip: When solving problems with parallel lines and transversals, always look for relationships such as corresponding angles, alternate angles, or co-interior angles to quickly determine unknown values.