Question

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

Short answer

No, the triangles are not congruent by SAS.

Step-by-step solution

Understand the SAS Congruence Criterion

The Side-Angle-Side (SAS) congruence criterion states that two triangles are congruent if two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle.

Identify the Given Congruence

From the problem, we have: \( \overline{AB} \cong \overline{DF} \), \( \overline{CA} \cong \overline{DE} \), and \( \angle C \cong \angle F \).

Analyze the Configuration

Check if the provided angle, \( \angle C \), is between the sides \( \overline{CA} \) and \( \overline{AB} \) in one triangle, and if \( \angle F \) is between \( \overline{DE} \) and \( \overline{DF} \) in the second triangle.

Determine Inclusion of the Angle

The given angles \( \angle C \) and \( \angle F \) are not between the given congruent sides in this case. We have \( \angle C \) between \( \overline{AC} \) and this would need to correspond to \( \angle D \) in the other triangle, if \( C \) and \( D \) are sections corresponding to corners.

Final Decision on Congruence

Since the given angle is not the included angle between the given side pairs in both triangles, the two triangles cannot be considered congruent by SAS.

Key concepts

SAS Criterion

The Side-Angle-Side (SAS) Criterion is fundamental in proving triangle congruence. It states that two triangles are congruent if two sides and the angle between them in one triangle are exactly equal to two sides and the included angle in another triangle.
If we denote the sides of a triangle as \( a, b \) and the angle between them as \( \theta \), for the SAS Criterion to be applicable, the corresponding sides in another triangle (also denoted as \( a, b \)) must have the same measurements, and the angle \( \theta \) must also match precisely.

• This is important because the angle specified must be the one formed by the two sides in question, creating a fixed shape which must match in both triangles.
• Unlike other congruence theorems, the placement of the angle is crucial here.

In practical geometry problems, always ensure that the given angle is the included angle to correctly apply the SAS Criterion. If the angles lie elsewhere, the criterion does not hold, and congruence cannot be concluded.

Congruent Triangles

Triangles are congruent when their corresponding sides and angles are equal. When two triangles are congruent, every angle and side in one triangle will exactly match a corresponding angle and side in the other.
Imagine two identical triangles cut from a piece of paper. If you place one over the other, they align perfectly; this is congruence at its core.

• Congruent triangles have many useful properties, such as they being able to be transformed into each other by rotations or reflections.
• Recognizing congruent triangles is valuable in solving problems, because it allows us to infer unknown measurements based on known corresponding parts.

There are several criteria to determine triangle congruence, such as SAS, SSS (Side-Side-Side), and ASA (Angle-Side-Angle). Each offers a unique way to establish that two triangles are congruent, even if not all sides or angles are immediately known.

Geometry Proofs

Geometry proofs are logical sequences used to demonstrate the truth of a geometric statement, often involving congruence or other properties of shapes.
A proof typically involves a series of well-reasoned steps that begin with given information and follow logical deductions to achieve a desired conclusion. In essence, proving geometry theorems is like following a recipe to ensure every ingredient (or premise) is used correctly.

• SAS is frequently used in geometry proofs to show the equality of triangles.
• Initiating a proof often includes marking given equalities or congruence directly onto diagrams, which can clarify relationships visually.

Creating a proof requires careful attention to which pieces of information are required, verifying each step builds on the last, and ensuring no steps are omitted. This makes geometric proofs rewarding yet challenging, as they solidify understanding and mastery of geometric concepts.