Given the figure, we need to find the measures of angles $a, b, c, d, e, f,$ and $g$ under the assumption that lines $L_1$ and $L_2$ are parallel, and $m$ is a transversal intersecting these parallel lines. We know that $\angle a = 23^\circ$.

Analysis

1. Corresponding Angles: When a transversal intersects two parallel lines, corresponding angles are equal. This property helps in deducing angles above $L_2$ and $L_1$.
2. Vertical Angles: Vertical angles are equal when two lines intersect. This property helps us find angles directly across from a given angle.
3. Supplementary Angles: Adjacent angles formed by a straight line sum up to $180^\circ$.

Solution

Let's use the properties systematically to determine each angle:

1. Angle $a$: $\angle a = 23^\circ \quad \text{(Given)}$

2. Angle $c$: $\angle c = \angle a = 23^\circ \quad \text{(Vertical angles)}$

3. Angle $b$: $\angle b = 180^\circ - \angle a = 180^\circ - 23^\circ = 157^\circ \quad \text{(Supplementary to } \angle a \text{)}$

4. Angle $d$: $\angle d = \angle b = 157^\circ \quad \text{(Corresponding angles)}$

5. Angle $g$: $\angle g = \angle c = 23^\circ \quad \text{(Corresponding angles)}$

6. Angle $f$: $\angle f = \angle a = 23^\circ \quad \text{(Vertical angles)}$

7. Angle $e$: $\angle e = \angle d = 157^\circ \quad \text{(Vertical angles)}$

Final Measures

• $\angle a = 23^\circ$
• $\angle b = 157^\circ$
• $\angle c = 23^\circ$
• $\angle d = 157^\circ$
• $\angle e = 157^\circ$
• $\angle f = 23^\circ$
• $\angle g = 23^\circ$

Let me know if you need further clarification on any of these steps or have questions about related topics.

Related Questions:

1. What are other pairs of corresponding angles in the diagram?
2. How would the angles change if $\angle a$ was different?
3. How do alternate interior angles behave with parallel lines?
4. What is the relationship between consecutive interior angles in parallel lines?
5. What happens to the angle measures if $L_1$ and $L_2$ are not parallel?

Tip: Always remember that corresponding angles are equal when a transversal intersects parallel lines!