Before learning the SSS formula let us recall what are congruence and similarity. If two triangles are congruent it means that three sides of one triangle will be (respectively) equal to the three sides of the other and three angles of one triangle will be (respectively) equal to the three angles of the other. If two triangles are similar it means that all corresponding angle pairs are equal and all corresponding sides are proportional. However, in order to be sure that the two triangles are similar or congruent, we do not necessarily need to have information about all sides and all angles. Let us understand the desired criterion using the SSS triangle formula using solved examples in the following sections.
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Using the SSS Formula, the congruency or similarity of any two triangles can be checked when two sides and the angle between these sides for both the triangles follow the required criterion. There are different SSS Triangle formulas used to prove the congruence or similarity between two triangles.
SSS Congruence Rule
The Side-Angle-Side theorem of congruency states that, if two sides and the angle formed by these two sides are equal to two sides and the included angle of another triangle, then these triangles are said to be congruent.
The SSS Similarity Rule
The SSS similarity criterion states that if the three sides of one triangle are respectively proportional to the three sides of another, then the two triangles are similar.
Let us see the applications of the SSS formula in the following solved examples section.
Solved Examples Using SSS Formula
Example 1: The two points P and Q are on the opposite sides of the line segment AB. The points P and Q are equidistant from points A and B. Can you prove that \(\Delta PAQ\) is congruent to the \(\Delta PBQ\)?
Solution: To prove: \(\Delta PAQ\) is congruent to the \(\Delta PBQ\)
Two points P and Q, equidistant from the endpoints of the line segment AB.
AP = BP
AQ = BQ
Now the side PQ is common in both the triangles \(\Delta PAQ\) and \(\Delta PBQ\).
Therefore according to the SSS Formula, the two triangles are congruent.
\[\Delta PAQ\cong\Delta PBQ\]
Answer: \(\Delta PAQ\cong\Delta PBQ\).
Example 2: Triangle ABC is an isosceles triangle and the line segment AD is the angle bisector of the angle A. Can you prove that \(\Delta ADB\) is congruent to the \(\Delta ADC\)?
Solution: To prove: \(\Delta ADB\) is congruent to the \(\Delta ADC\)
Given: An isosceles triangle, ABC
Therefore, AB = AC.
Now the side AD is common in both the triangles \(\Delta ADB\) and \(\Delta ADC\).
As the line segment AD is the angle bisector of the angle A then it divides the line segment BC into two equal parts BD and CD.
BD = CD
AB = AC
Now according to the SSS formula, the two triangles are congruent.