Triangle A B D is inside triangle A C E. Angles A B D and C are congruent. Angles A D B and E are congruent.

Question

A classmate says that because the angle ABD is congruent to the angle ACE and the angle ADB is congruent to the angle AEC,BD is congruent CE to How do you respond?

Answer 1

To respond to your classmate's claim about the triangles, we need to analyze the information given and apply the principles of triangle congruence.

You mention that triangle ABD is inside triangle ACE, and that angles ABD and ACE are congruent, as well as angles ADB and AEC. This information suggests that the two triangles share certain angle properties.

However, knowing that two angles are congruent in each triangle does not automatically imply that the sides opposite those angles are congruent. For two triangles to be congruent (and therefore have corresponding sides that are also congruent), you need to establish a clear congruence condition.

In this case, we have established the following:

1. \( \angle ABD \cong \angle ACE \)
2. \( \angle ADB \cong \angle AEC \)

Since you have two pairs of congruent angles, it appears you are on the right track to consider triangle congruence. However, we need a third angle to use the Angle-Angle-Angle (AAA) similarity criterion or another congruence theorem (like Angle-Side-Angle or Angle-Angle-Side).

In triangle situations, we generally apply the following rules:

• Angle-Angle (AA) Similarity: If two angles of one triangle are congruent to two angles of another triangle, the two triangles are similar, which means their corresponding sides are proportional, but not necessarily equal.
• Angle-Side-Angle (ASA) or Side-Angle-Side (SAS) congruence: If we have the corresponding sides or sides included between the angles, we can establish congruence.

In conclusion, based on what your classmate presented, the triangles ABD and ACE may be similar (if all angles are considered), but you cannot conclude that sides BD and CE are congruent without more information, such as knowing the lengths of the sides or having an additional congruence condition.

You could respond with:

"While you’ve correctly identified the angle congruences, we need to apply triangle congruence criteria to say that BD is congruent to CE. The triangles ABD and ACE might be similar due to the angle congruences, but that doesn't guarantee that the sides are equal without additional information."