SAS Triangle Formula

Before learning the SAS triangle formula let us recall what is congruence and similarity. Congruence of triangles means that all corresponding angle pairs are equal and all corresponding sides are equal. 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. Using SAS Triangle Formula, 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. Let us understand the desired criterion using the SAS triangle formula using solved examples in the following sections.

What Is SAS Triangle Formula?

There are different SAS Triangle formulas used to prove the congruence or similarity between two triangles. SAS stands for Side-Angle-Side.

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 SAS Similarity Rule

The SAS similarity criterion states that If two sides of one triangle are respectively proportional to two corresponding sides of another, and if the included angles are equal, then the two triangles are similar.

Let us see the applications of SAS triangle formulas in the following section.

Examples Using SAS Triangle Formula

Example 1: James wanted to know which congruency rule says that these triangles are congruent. Let's help him figure this out.

Solution:     

To find: The congruency rule followed for given case

Given:

EF = MO = 3 in

FG = NO = 4.5 

\( \angle{EFG} = \angle{MON} =110^{\circ}\)

\(\triangle{EFG} \cong \triangle{MNO} (\text{By SAS rule})\)

Answer: These triangles are congruent by the SAS triangle formula.

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 \(\Delta ADC\) by using SAS triangle formula? What do you know about BD and CD?

Solution:       

To Prove: \(\Delta ADB\) is congruent to the \(\Delta ADC\)

Given:

The triangle ABC is an isosceles triangle, with 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 \(\angle A\) into two equal parts.

Therefore, \(\angle BAD=\angle CAD\)

Now according to the SAS rule, the two triangles are congruent.

Hence,

\(\Delta ADB \cong \Delta ADC\)

Therefore, the other side and the other two angles are bound to be congruent. BD = CD                                                               

Answer: \(\Delta ADB \cong \Delta ADC\) and BD = CD