Two congruent triangles are defined to be congruent when they are of the same shape and size. Congruence of two triangles can be checked by some rules or postulates. This is the most likely to be the most common one and it is occasionally referred to as Side-Angle-Side, or SAS Congruence Postulate.
A lot of geometrical assessment and practice activities contain the following question:
Which pair of triangles can be proven congruent by SAS?
You need to look at triangles in pairs and to check that they are obeying the rules as laid down by the SAS Postulate. SAS congruence criterion In the event that SAS is equal, i.e. two sides of a triangle and the angle between them are equal to two sides of another triangle and the angle between them, then the two triangles will be congruent.
A diagram may be presented to you of two triangles – say △ABC and △DEF where:
AB ≅ DE
- AC ≅ DF
- ∠A ≅ ∠D
This example reveals how SAS is applied when checking congruence, though the angle considered should fall inside the sides and not outside them.
Explanation
To get a clearer picture of how the SAS postulate works, we should look at the basic idea of triangle congruence.
What Does Triangle Congruence Mean?
Two triangles are congruent if both their lengths and angles are only parallel with one another. That is:
- The three sides of one triangle have the same length as the sides of the other.
- The triangle has three equal sides that measure to the corresponding angles of the other.
There are several rules for triangle congruence, including:
- SSS (Side-Side-Side)
- ASA (Angle-Side-Angle)
- AAS (Angle-Angle-Side)
- SAS (Side-Angle-Side)
- HL (Hypotenuse-Leg for right triangles)
What is the SAS Congruence Postulate?
SAS Postulate states:
When two sides of one triangle and the angle between them are congruent to two sides of another triangle and the angle between them then the triangles are congruent.
This postulate is based on a certain ordering:
The angle should come between the two equal sides, which are known. When the angle is on the opposite side, then congruence (in SSA) cannot always confirm.
We will explain this by a specific example:
In △XYZ and △PQR, suppose we are given:
- XY = PQ
- YZ = QR
- ∠Y = ∠Q
Angle ∠Y lies between sides XY and YZ and sides XY and YZ are respectively equal to sides PQ and QR, then we can say:
△XYZ ≅ △PQR by SAS
Theoretical geometry and practical professions such as construction, architecture, and engineering benefit a lot from this postulate.
End Note
Being able to use SAS Postulate means you are well on the way to mastering geometry. It checks if two triangles are the same only by seeing two sides and the angle between them. Nevertheless, make sure you handle your data carefully, since not paying attention to the angle or making the wrong guess about congruence may result in wrong answers.
When someone asks about the SAS congruence rule, always go over the requirements, look at your diagram, and use the rule. No matter if the application is in exam situations, applied to real-life problems, or in technical calculations, learning this postulate is very important.
FAQs
Q1: What does SAS stand for in triangle congruence?
SAS means two sides and the included angle of a triangle will be congruent to the matching parts of another triangle.
Q2: Can triangles be congruent by SSA?
You cannot use SSA (Side-Side-Angle) to prove that double triangles stay congruent, as sometimes it leads to multiple results.
Q3: Why is the included angle important in SAS?
The angle guarantees the two sides have a unique connection, which can result in a definite triangle shape.
Q4: What should I look for in a diagram to confirm SAS?
Make sure each of the triangles has two paired identical sides and one identical angle between them.
