Question

Use the given information to determine whether the two triangles are congruent by SAS. Write yes or no. $$\overline{E F} \cong \overline{C A}, \overline{B C} \cong \overline{E D}, \angle C \cong \angle E$$

Short answer

No, the triangles are not congruent by SAS.

Step-by-step solution

Identify Given Information

The problem provides the following congruence information between the triangles: \(\overline{EF} \cong \overline{CA}\), \(\overline{BC} \cong \overline{ED}\), and \(\angle C \cong \angle E\).

Understand SAS Congruence Criterion

The Side-Angle-Side (SAS) theorem states that two triangles are congruent if two sides and the angle between them are respectively equal in both triangles.

Check Sides and Included Angle

Examine the given congruent sides and angle: \(\overline{BC} \cong \overline{ED}\) and \(\overline{EF} \cong \overline{CA}\) are provided, along with \(\angle C \cong \angle E\). However, the angle \(\angle C\) given as congruent is not the included angle between the two sides \(\overline{BC}\) and \(\overline{CA}\) in triangle \(\triangle ABC\), nor between \(\overline{ED}\) and \(\overline{EF}\) in triangle \(\triangle DEF\).

Determine Congruence Status

Since the congruent angle is not between the given sides, we cannot determine that the triangles are congruent by SAS criterion.

Key concepts

Triangle Congruence

Triangle congruence is a fundamental concept in geometry. It means that two triangles are exactly the same size and shape, allowing them to be perfectly overlapped.
Understanding when two triangles are congruent is a vital skill in geometry, as it can help prove other geometrical properties and solve complex problems more easily.

• Congruent triangles have equal corresponding sides and angles.
• They often appear in various forms in geometry, so recognizing congruence criteria is essential.
• Not all triangles that look similar are congruent; specific criteria must be met.

To establish triangle congruence, several methods, or criteria, can be used, such as SAS (Side-Angle-Side), ASA (Angle-Side-Angle), and SSS (Side-Side-Side).
Each criterion involves specific angle and side relationships that, when met, confirm congruency.

Geometry Problem-Solving

Geometry problem-solving often involves understanding properties of shapes and using congruence to determine angles, side lengths, and other attributes.
In problems similar to the presented exercise, steps to solve such include:

• Identifying given information: This includes recognizing known sides and angles.
• Applying relevant theorems: These might be congruence or other geometric principles.
• Checking conditions met by congruence criteria: Ensure any stated criteria are fully met.

Engaging with geometry problems in this structured method assists in avoiding errors.
By following a systematic approach, you can verify what you know and use logic to deduce unknown information.

Congruence Criteria

Congruence criteria are specific rules that determine when two triangles are congruent.
Familiarizing with these rules is key:

• SAS (Side-Angle-Side): Two triangles are congruent if two sides and the included angle are equal.
• ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) ensure congruency if two angles and the connecting side (in ASA) or a non-included side (in AAS) are equal.
• SSS (Side-Side-Side): All three sides are equal across both triangles.

Each of these ensures the two triangles will be mirror images concerning size and shape.
To use the SAS criterion properly, the angle must be between the two sides, as only this assures the triangles are congruent.