The Side-Angle-Side (SAS) Congruence Theorem: Proving Triangles Congruent.

If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

An application of the Side-Angle-Side (SAS) Congruence Theorem

An application of the Side-Angle-Side (SAS) Congruence Theorem. According to this theorem, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent.

To understand why this is true, let’s break it down step by step:

1. Two sides: When we say that two sides are congruent, it means that they have the same length. In our case, we are given that two sides of one triangle are congruent to two sides of another triangle.

2. Included angle: The included angle is the angle between the two given sides. In our case, we are given that the included angles of both triangles are congruent.

Now, let’s assume we have triangle ABC and triangle DEF. We are given that side AB is congruent to side DE, side AC is congruent to side DF, and angle BAC is congruent to angle EDF. We need to prove that triangle ABC is congruent to triangle DEF.

To prove this, we can use the SAS Congruence Theorem as follows:

1. Side AB is congruent to side DE (given).
2. Side AC is congruent to side DF (given).
3. Angle BAC is congruent to angle EDF (given).

By using these three pieces of information, we can conclude that triangle ABC is congruent to triangle DEF. This can be written as:

∆ABC ≅ ∆DEF (using the congruence symbol ≅).

So, in summary, if two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the triangles are congruent according to the Side-Angle-Side (SAS) Congruence Theorem.

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