Question

A sequence of rigid transformations maps △ABC→△A′′B′′C′′ . Given ∠C≅∠C′′ and BC¯¯¯¯¯¯¯¯≅B′′C′′¯¯¯¯¯¯¯¯¯¯¯¯ , which additional congruence statement would be needed to prove the triangles are congruent by the ASA Congruence Theorem? Option #1: ∠A≅∠A′′ Option #2: ∠B≅∠B′′ Option #3: AB¯¯¯¯¯¯¯¯≅A′′B′′¯¯¯¯¯¯¯¯¯¯¯¯ Option #4: AC¯¯¯¯¯¯¯¯≅A′′C′′¯¯¯¯¯¯¯¯¯¯¯¯ (1 point) Option #​ would be needed to prove the triangles are congruent by the ASA Congruence Theorem.

Answer 1

To prove that triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) are congruent using the ASA (Angle-Side-Angle) Congruence Theorem, we need two angles and the included side of one triangle to be congruent to the corresponding parts of the other triangle.

You already have:
- \( \angle C \cong \angle C'' \) (one angle is provided)
- \( BC \cong B''C'' \) (the side between the two angles is provided)

Now, you need one more angle, which would be either \( \angle A \) or \( \angle B \), to complete the ASA condition.

Then, the options for the additional congruence statements are:

1. Option #1: \( \angle A \cong \angle A'' \)
2. Option #2: \( \angle B \cong \angle B'' \)
3. Option #3: \( AB \cong A''B'' \)
4. Option #4: \( AC \cong A''C'' \)

Since choices #3 and #4 provide sides, they won't help in establishing the required angle-side-angle congruence. Therefore, you need either \( \angle A \) or \( \angle B \) to be congruent. Thus, the correct additional congruence statement would be either:

- Option #1: \( \angle A \cong \angle A'' \) or
- Option #2: \( \angle B \cong \angle B'' \)

So, the answer is either Option #1 or Option #2.