Final answer:
The proof involves understanding that the reflection of triangle ABC to A1B1C1 over BC makes the two triangles congruent. Congruency ensures that corresponding angles are equal, thus making the ray the angle bisector of angle ABA1 by definition.
Step-by-step explanation:
The question requires proof that the ray is the angle bisector of angle ABA1, in the context of a triangle ABC and its reflected image A1B1C1 across line BC. By the definition of reflection, line BC will act as a mirror and therefore, triangle ABC is congruent to triangle A1B1C1. Since congruent triangles have all corresponding angles equal, the angle ABC is equal to angle A1B1C. Additionally, because the reflection produces two identical but opposite triangles, angle ABA1 is indeed bisected by the line of reflection, making the ray the angle bisector of angle ABA1.
When a ray of light approaches a reflective surface such as a mirror, the angle of incidence (the angle the ray makes with a line perpendicular to the surface at the point of incidence) is equal to the angle of reflection. Using this physical principle, we can establish that the angles created by the intersection of a ray and the line of reflection are congruent. Therefore, the reflected ray from the surface at angle ABA1 is proof that the ray is the angle bisector.
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