Question

If \(\triangle A B C \cong \triangle D E F\), name the corresponding side or angle. $$\overline{A C}$$

Short answer

The corresponding side is \(\overline{DF}\).

Step-by-step solution

Understand Congruent Triangles

When two triangles are congruent, all corresponding sides and angles are equal. This means that if triangle ABC is congruent to triangle DEF, then their corresponding parts are identical.

Identify Corresponding Parts

To find the part of triangle DEF that corresponds to the side \(\overline{AC}\) in triangle ABC, we look at the order of the vertices in the congruence statement \(\triangle ABC \cong \triangle DEF\). The order of the letters tells us which parts correspond. Here, \(A\) corresponds to \(D\), \(B\) corresponds to \(E\), and \(C\) corresponds to \(F\).

Match Corresponding Side

Based on the ordering from congruence \(\triangle ABC \cong \triangle DEF\), the side \(\overline{AC}\) in triangle ABC corresponds to the side \(\overline{DF}\) in triangle DEF.

Key concepts

Triangle Correspondence

When discussing congruent triangles, one of the first things to understand is "triangle correspondence." Triangle correspondence is the order in which the vertices of two triangles are matched up. This order helps you identify which parts of one triangle are equal to which parts of another.

For example, if you have the notation \( \triangle ABC \cong \triangle DEF \), the letters indicate how the triangles correspond:

• The vertex \( A \) corresponds with vertex \( D \).
• The vertex \( B \) corresponds with vertex \( E \).
• The vertex \( C \) corresponds with vertex \( F \).

Understanding how each vertex matches helps you find the equivalent sides and angles between the triangles. This is crucial in any problems involving congruent triangles, as it lays the foundation for identifying all congruent parts.

Triangle Congruence

Triangle congruence is a concept where two triangles are said to be congruent if all their corresponding sides and angles are identical. The notation \( \triangle ABC \cong \triangle DEF \) serves as a declaration that all these corresponding parts are equal.

Congruence can be established through certain properties or criteria like SSS (Side-Side-Side), SAS (Side-Angle-Side), or ASA (Angle-Side-Angle). These methods allow one to show two triangles are "mirror images" of each other in terms of size and shape, even though their orientations might differ.

Whenever you see the congruence symbol \( \cong \), it is a signal that every pair of sides and angles in the two triangles match perfectly. This knowledge allows you to address specific problems by simply reading the order of the triangle correspondence, knowing the parts will be congruent.

Corresponding Sides

In the context of congruent triangles, corresponding sides are the sides that are in the same position relative to each vertex correspondence. With \( \triangle ABC \cong \triangle DEF \), if you are looking at side \( \overline{AC} \), you need to identify which side it matches in \( \triangle DEF \).

According to the correspondence:

• Side \( \overline{AC} \) corresponds to side \( \overline{DF} \).

This means that the length of \( \overline{AC} \) in \( \triangle ABC \) is equal to the length of \( \overline{DF} \) in \( \triangle DEF \). This principle is fundamental when working out problems that involve calculating angles, side lengths, or proving additional geometric properties based on the given congruence. Therefore, understanding the concept of corresponding sides is vital to solving geometric problems correctly.