Question

The ASA Congruence Theorem Practice
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Question
Use the image to answer the 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

it was option 2

Answer 2

To prove that triangles \( \triangle ABC \) and \( \triangle A''B''C'' \) are congruent by the ASA (Angle-Side-Angle) Congruence Theorem, we already have:

1. \( \angle C \cong \angle C'' \) (given),
2. \( BC \cong B''C'' \) (given).

We need one more angle to complete the ASA requirement.

Given the options, we need to identify the additional angle that we can show is congruent:

- Option #1: \( \angle A \cong \angle A'' \)
- Option #2: \( \angle B \cong \angle B'' \)
- Option #3: \( AB \cong A''B'' \) (which is a side, not an angle)
- Option #4: \( AC \cong A''C'' \) (which is also a side, not an angle)

Since we're looking for an angle to complete the ASA congruence, we can see that options #3 and #4 do not fulfill the requirement since they provide side congruence, not angle congruence.

Hence, we need to choose either Option #1 or Option #2, which both give us the necessary angle congruence.

Since the problem does not specify which angle is needed, you could choose either:

**Therefore, Option #1 or Option #2 would be needed to prove the triangles are congruent by the ASA Congruence Theorem.**

If you have to pick only one and if there are no further specifications, either is valid, but traditionally you might want to choose one based on the context, so:

**Option #1: \( \angle A \cong \angle A'' \) could be a good choice, but Option #2: \( \angle B \cong \angle B'' \) would also work.**

Answer 3

Thank you for the clarification! So, the correct additional congruence statement required to prove the triangles are congruent by the ASA Congruence Theorem is:

**Option #2: \( \angle B \cong \angle B'' \)**

With \( \angle C \cong \angle C'' \), \( BC \cong B''C'' \), and the additional statement \( \angle B \cong \angle B'' \), it satisfies the ASA criteria for congruence.

If you have any more questions or need further assistance, feel free to ask!