One of the most common methods of proving the congruence of triangles is the Side-Angle-Side (SAS) theorem. This theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. The SAS theorem is a powerful tool for proving the congruence of triangles and is widely used in geometry and trigonometry. In this article, we will discuss the concept of congruence and how it applies to the SAS theorem.

What is Congruence?

Congruence is an important concept in geometry and trigonometry and is used to prove the equality of two shapes. When two shapes are congruent, they will have the same size and shape. This means that all of their sides and angles will be equal. In other words, if two triangles are congruent, then they will have the same size and shape, and all three sides and angles of each triangle will be equal.

What is the SAS Theorem?

The SAS theorem is a powerful tool for proving the congruence of two triangles. This theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. The included angle is the angle between the two sides that are being compared. For example, if sides AB and BC of triangle ABC are congruent to sides XY and YZ of triangle XYZ, then angle BAC is the included angle.

How to Prove the Congruence of Two Triangles by SAS

When proving the congruence of two triangles using the SAS theorem, you must first determine if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. If this is the case, then the two triangles are congruent. To do this, you must calculate the lengths of the sides and the measure of the included angle of each triangle. Once you have determined that the two sets of sides and angles are congruent, then you can prove that the two triangles are congruent.

Examples of Proving Congruence of Triangles by SAS

To illustrate how the SAS theorem is used to prove the congruence of two triangles, let\’s take a look at a few examples. In each example, we will use the given measurements of the sides and angles to determine if the SAS theorem applies.

Example 1

In this example, we have two triangles, ABC and XYZ. The sides and angles of triangle ABC are given as follows: AB = 4 cm, BC = 5 cm, and angle BAC = 90°. The sides and angles of triangle XYZ are given as follows: XY = 4 cm, YZ = 5 cm, and angle YXZ = 90°. Since the sides and angles of both triangles are equal, we can use the SAS theorem to prove that the two triangles are congruent.

Example 2

In this example, we have two triangles, ABC and XYZ. The sides and angles of triangle ABC are given as follows: AB = 7 cm, BC = 3 cm, and angle BAC = 60°. The sides and angles of triangle XYZ are given as follows: XY = 7 cm, YZ = 3 cm, and angle YXZ = 60°. Since the sides and angles of both triangles are equal, we can use the SAS theorem to prove that the two triangles are congruent.

Conclusion

The Side-Angle-Side (SAS) theorem is a powerful tool for proving the congruence of two triangles. The theorem states that if two sides and the included angle of one triangle are congruent to two sides and the included angle of a second triangle, then the two triangles are congruent. To prove the congruence of two triangles using the SAS theorem, you must first determine if two sides and the included angle of one triangle are congruent to two sides and the included angle of the other triangle. If this is the case, then the two triangles are congruent. Finally, you must use the given measurements of the sides and angles to determine if the SAS theorem applies.