1. **Statement of the problem:** Given two intersecting lines with a horizontal line passing through points A, B, and C (B below segment AC), and a diagonal line crossing near C forming an angle of 126° at point C with the horizontal line, find the measure $m\angle B$ in degrees. 2. **Analyzing the geometry:** The angle of 126° is formed at point C between the horizontal line and the diagonal line. Since the horizontal line is straight, the angle adjacent to 126° on the same straight line is its supplementary angle. 3. **Calculating the supplementary angle:** The adjacent angle at point C on the horizontal line is $$180° - 126° = 54°.$$ 4. **Identifying $m\angle B$:** Because lines intersect, vertical angles are equal. $m\angle B$ is the vertical angle opposite to the 54° angle at C, so $$m\angle B = 54°.$$ 5. **Final answer:** $$m\angle B = 54°.$$