Side-Angle-Side (SAS)
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.
The Side-Angle-Side (SAS) is a theorem used in geometry to prove that two triangles are congruent. According to the 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 two triangles are congruent. In other words:
– Given two triangles ABC and DEF, if AB = DE, AC = DF, and angle A = angle D, then triangle ABC is congruent to triangle DEF.
To prove this theorem, we can use the following steps:
1. Draw two triangles ABC and DEF.
2. Label the sides and angles as shown in the theorem.
3. Use the congruence of the two sides and the included angle to prove that the triangles have one matching side-angle-side.
4. Use the congruence of the remaining sides and angles to prove that the triangles have another matching side-angle-side.
5. Conclude that the two triangles are congruent by the SAS theorem.
It is important to note that the order of the sides and angle in the theorem matters. If the side-angle-side are not in the same order in both triangles, then we cannot conclude that they are congruent.
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