Question

Three triangles labeled upper A upper B upper C, upper A prime upper B prime upper C prime, and upper A double prime upper B double prime upper C double prime intersect at the point of C, where upper C prime and upper C double prime are also located.
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 by the ASA (Angle-Side-Angle) Congruence Theorem, we already have the following:

1. \( \angle C \cong \angle C'' \) (the angles at point C are congruent)
2. \( \overline{BC} \cong \overline{B''C''} \) (the sides opposite to angles A and A'' are congruent)

We need one additional angle congruence statement to satisfy the ASA condition, which requires two angles and the included side (the side that is between the two angles).

Now, let's analyze the options:

- **Option #1: \( \angle A \cong \angle A'' \)** - This would mean that we have angles A and A'' with the included side \( \overline{BC} \), which would satisfy the ASA condition.

- **Option #2: \( \angle B \cong \angle B'' \)** - This would also provide an included angle with side \( \overline{BC} \), satisfying the ASA condition but focusing on angle B instead.

- **Option #3: \( \overline{AB} \cong \overline{A''B''} \)** - This is a side congruence statement and does not help us complete the ASA condition.

- **Option #4: \( \overline{AC} \cong \overline{A''C''} \)** - Like Option #3, this is also a side congruence statement, which does not fulfill the ASA condition.

To prove the triangles are congruent using the ASA Congruence Theorem, we need an additional angle congruence. The additional statement could be any of the first two options.

In this specific context, either option would technically suffice, but since they are asking for one specific option:

You could choose **Option #1: \( \angle A \cong \angle A'' \)** or **Option #2: \( \angle B \cong \angle B'' \)**.

However, if I must pick one for this particular response, I will select:

**Option #1: \( \angle A \cong \angle A'' \)**.