Question

Triangle A′B′ C' is a reflection of ABC across line BC. Prove that ray BC is the angle bisector of angle AB′ A' .

a. Prove that BC is perpendicular to A'B′.
b. Prove that AB ′ A′ ≅AB ' C′.
c. Prove that BC bisects angle AB ′A ′.
d. Prove that A ′ B ′≅A 'C'

Answer 1

To prove that ray BC is the angle bisector of angle AB' A', we can use the laws of reflection and congruence. BC is perpendicular to A'B', AB' A' is congruent to AB ' C', BC bisects angle AB' A', and A' B' is congruent to A 'C'.

Step-by-step explanation:

To prove that ray BC is the angle bisector of angle AB' A', we need to show that the angle of incidence and the angle of reflection are equal. If triangle ABC is reflected across line BC to form triangle A'B'C', the angles of incidence and reflection will be equal since the law of reflection states that the angle of incidence is equal to the angle of reflection.

1. To prove that BC is perpendicular to A'B', we can use the fact that in a reflection, lines that are perpendicular in the original shape will remain perpendicular in the reflected shape. Since BC is perpendicular to AB in triangle ABC, BC will still be perpendicular to A'B' in triangle A'B'C'.
2. To prove that AB' A' ≅AB ' C', we can use the fact that a reflection creates congruent shapes. Triangle ABC is congruent to triangle A'B'C' because they are reflections of each other across line BC. Therefore, AB' A' is congruent to AB ' C'.
3. To prove that BC bisects angle AB' A', we can use the fact that lines that are bisectors in the original shape will remain bisectors in the reflected shape. Since BC bisects angle ABC in triangle ABC, BC will also bisect angle A'B' A' in triangle A'B'C'.
4. To prove that A' B'≅A 'C', we can use the fact that congruent shapes in the original shape will remain congruent in the reflected shape. Since AB is congruent to A'B' in triangle ABC, AB will remain congruent to A' B' in triangle A'B'C'.