If two triangles have three congruent, corresponding angles, what additional information is needed to prove that the triangles are congruent?

General
1

if two triangles have three congruent, corresponding angles, what additional information is needed to prove that the triangles are congruent?

2

LectureNotes said if two triangles have three congruent, corresponding angles, what additional information is needed to prove that the triangles are congruent?

Answer:

To determine if two triangles are congruent, having three corresponding congruent angles (which makes them similar) is not sufficient. We need additional information about the sides of the triangles. Here’s a detailed explanation of what is required:

  1. Side Lengths Correspondence:
    • Angle-Angle-Side (AAS) Congruence Theorem:
      • If two triangles have two angles and the corresponding non-included side congruent, then the triangles are congruent.
    • Angle-Side-Angle (ASA) Congruence Theorem:
      • If two triangles have two angles and the included side congruent, then the triangles are congruent.
    • Side-Angle-Side (SAS) Congruence Theorem:
      • If two triangles have two corresponding sides and the included angle congruent, then the triangles are congruent.
    • Side-Side-Side (SSS) Congruence Theorem:
      • If all three corresponding sides of the triangles are congruent, then the triangles are congruent.

Detailed Example:

Suppose we have two triangles, \triangle ABC and \triangle DEF. We are given that:

  • \angle A \cong \angle D
  • \angle B \cong \angle E
  • \angle C \cong \angle F

This makes \triangle ABC \sim \triangle DEF (the triangles are similar), but not necessarily congruent. To show congruence, we need to add information about the sides.

  1. Using AAS:

    • Suppose we know side BC \cong EF in addition to the angles matching.
    • Hence, \triangle ABC and \triangle DEF are congruent by AAS.

  2. Using ASA:

    • Suppose we know side AB \cong DE, which is between \angle A and \angle B (and respectively \angle D and \angle E).
    • Hence, \triangle ABC and \triangle DEF are congruent by ASA.

  3. Using SAS:

    • Suppose we know two sides such as AB \cong DE and AC \cong DF, and the included angle \angle A \cong \angle D.
    • Hence, \triangle ABC and \triangle DEF are congruent by SAS.

  4. Using SSS:

    • Suppose we know AB \cong DE, BC \cong EF, and AC \cong DF.
    • Hence, \triangle ABC and \triangle DEF are congruent by SSS.

Final Answer:
For two triangles having three congruent, corresponding angles to be congruent, we need to confirm the congruence of at least one pair of corresponding sides. Possible congruence criteria include AAS, ASA, SAS, or SSS.