To solve for $m\angle B$, let's break down the problem step by step.

Given:

• $\overline{AB} \parallel \overline{DF}$ (AB is parallel to DF),
• $m\angle DCE = 105^\circ$,
• $m\angle CEF = 150^\circ$.

Since $\overline{AB} \parallel \overline{DF}$, we know that the angles involving these lines might be related through corresponding angles, alternate interior angles, or by applying other geometric properties.

Step 1: Analyze $\triangle CEF$

From the given, we have $m\angle CEF = 150^\circ$. In a triangle, the sum of the internal angles is always 180 degrees. So, using the fact that the triangle has an angle of 150 degrees at $E$, the other two angles must sum to $30^\circ$.

Let's first look at the relationship between angles formed by the parallel lines:

Step 2: Use parallel line properties

Since $\overline{AB} \parallel \overline{DF}$ and we're dealing with a transversal through $\overline{AB}$, you can use alternate interior or corresponding angles to relate $\angle B$ to the angles on line DF. Based on the geometry setup, $m\angle B$ is complementary to one of these angles.

Solution:

Now, apply these findings to the problem:

Let me know if you need any further details or clarification about the steps!

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Here are 5 questions to expand on this topic:

1. How can you use corresponding angles to solve parallel line problems?
2. What are the key angle properties of a triangle?
3. How do you solve for an unknown angle in a transversal crossing parallel lines?
4. How do alternate interior angles help in parallel line problems?
5. How do supplementary and complementary angles differ in parallel line scenarios?

Tip: In any geometry problem, always look for parallel lines, as they often unlock key angle relationships like corresponding and alternate interior angles!