https://file.aiquickdraw.com/mathbot/file/1732175894016lq90j4x9.jpg

From the given diagram, we see two congruent triangles \( \triangle ABC \) and \( \triangle DEF \). The given information includes:

1. \( \angle BAC = \angle EDF \) (corresponding angles are equal).
2. \( AB = DE \) (corresponding sides are equal).
3. \( AC = DF \) (corresponding sides are equal).

To determine the condition that validates their congruence, we apply the criteria for triangle congruence:

### Congruence Criteria:
1. **SAS (Side-Angle-Side)**: If two sides and the included angle of one triangle are equal to the corresponding sides and included angle of another triangle, the triangles are congruent.
   
   - Here, \( AB = DE \), \( AC = DF \), and \( \angle BAC = \angle EDF \). The angle \( \angle BAC \) is included between the sides \( AB \) and \( AC \), satisfying the SAS criterion.
   
Thus, the triangles \( \triangle ABC \) and \( \triangle DEF \) are congruent by **SAS criterion**.

Let me know if you'd like a deeper explanation of congruence or have further questions! Here are some relative questions:

1. What is the difference between SAS and other congruence criteria?
2. Can we prove these triangles congruent by another criterion like SSS or ASA?
3. Why is the inclusion of the angle important for SAS congruence?
4. How can congruence be used in practical geometry problems?
5. Are there scenarios where \( \triangle ABC \) and \( \triangle DEF \) are not congruent?

**Tip:** Always check whether the included angle matches the sides specified when applying SAS!

Triangles ABC and DEF are congruent, given ∠BAC = ∠EDF, AB = DE, and AC = DF. Which of the given conditions is valid?

Explore how triangles ABC and DEF are proven congruent using the SAS criterion, given ∠BAC = ∠EDF, AB = DE, and AC = DF. This step-by-step solution demonstrates the application of congruence in geometry for students in Grades 6-8.

Math Problem Statement

Solution

Ask a new question for Free

By Image

Math Problem Analysis

Mathematical Concepts

Formulas

Theorems

Suitable Grade Level

Related Recommendation