To solve for the measure of $\angle B$, let’s analyze the given information and apply geometric principles:

1. Given Information:

   • $\overline{AB}$ is congruent to $\overline{BC}$, which implies $\triangle ABC$ is isosceles with $AB = BC$.
   • $\overline{AE}$ intersects $\overline{BF}$ at point $C$.
   • The measure of $\angle ECF$ is $26^\circ$.

2. Objective:

   • Find the measure of $\angle B$.

3. Solution Approach: Since $\triangle ABC$ is isosceles with $AB = BC$, the base angles of the triangle, $\angle BAC$ and $\angle BCA$, are equal. Let’s denote the measure of each base angle by $x$.

4. Using the Triangle Sum Theorem: In any triangle, the sum of the angles is $180^\circ$. Therefore: $\angle B + 2x = 180^\circ$

5. Vertical Angles: The $26^\circ$ angle at point $C$ implies that $\angle BCA = 26^\circ$ as well due to the linear pair formed by the intersecting lines at point $C$. Hence, we have: $x = 26^\circ$

6. Calculating $\angle B$: Substitute $x = 26^\circ$ into the equation: $\angle B + 2(26^\circ) = 180^\circ$ $\angle B + 52^\circ = 180^\circ$ $\angle B = 180^\circ - 52^\circ$ $\angle B = 128^\circ$

Answer:

The measure of $\angle B$ is $128^\circ$.

Would you like more details or have any questions?

Here are some related questions to expand your understanding:

1. How do you determine the base angles in an isosceles triangle?
2. Why does the sum of angles in a triangle always equal $180^\circ$?
3. What are vertical angles, and why are they always equal?
4. How do intersecting lines affect angle relationships in a geometric figure?
5. Can an isosceles triangle have an angle measure greater than $128^\circ$?

Tip: When solving geometry problems, always look for congruent lines or angles as they can reveal properties like symmetry or angle relationships.