Difference Between ASA and AAS

ASA and AAS are two of the methods for proving congruency among triangles. ASA stands for Angle, Side, Angle whereas AAS stands for Angle, Angle, Side. ASA aligns to the congruency associated with an included side and any two angles. AAS aligns to the congruency associated with a non-included side and two corresponding angles.

ASA vs AAS

The main difference between ASA and AAS is that ASA is easier for being used to prove congruency rather than the AAS congruency. ASA is the origination of angles with the help of two lines comprising non-included angles and the same transversal. Whereas, AAS is the origination of angles by two lines with the help of included angle and the same transversal.

ASA vs AAS

In ASA, the requirement that the triangles are congruent is satisfied if the vertices of two triangles are in one-to-one correspondence in a way such as the two angles and the included side of one triangle are congruent to the two angles and the included side of the second triangles, respectively.

AAS or angle, angle, side congruency is associated with non-vertex angles. It cannot be used for identifying the degree of similarity. Algebraic manipulation cannot be used during this congruency as it is based on two pairs of angles that are similar to each other. It involves two lines intersecting each other.

Comparative Table Between ASA and AAS

Parameters of comparison

ASAAAS
Abbreviation
The abbreviation of ASA is “Angle, Side, Angle”. It indicates the incorporation of both angles and the side that is included.
The abbreviation of AAS is “Angle, Angle, Side”. It indicates the incorporation of corresponding two angles and a side that is non-included.
Definition
ASA indicates the congruency established in two triangles having equal sides in between equal angles that are corresponding.
Congruency is established when the two angles and opposite sides of them are congruent to angles corresponding to an independent side of another triangle.
Inclusion of side
Unlike AAS congruency, the representation of “Angle, Angle, Side” has the involvement of side in its representation of postulate.
Unlike, ASA congruency, the representation of “Angle, Side, Angle” has the involvement of the side in its representation of postulate.
Proof
ASA can be referred to as a prove for congruency. It uses geometry for proving its congruency but not trigonometry.
AAS can be referred to as a prove for similarity. It uses trigonometry as well as geometry for proving its congruence.
Other definition
It can also be defined as the formation of angles by both lines involving non-included angles and the same transversal.
It can also be defined as the formation of angles by both lines involving an included angle and the same transversal.

What is ASA?

Two triangles are said to be congruent to each other when both the triangles contain an equal side incorporated among equal angles that are corresponding to one another. When the vertices among two triangles have correspondence in one-to-one such as two angles along with the side included in one of the triangles are respectively congruent to both the angles and the side included in another triangle.

This very situation proves that both the triangles are congruent to each other. Both the triangles are proved to be congruent when the included side and two angles of two triangles are equal to each other. It is associated with the formula, A=B-C. the value associated with the congruency ranges from 0 degrees to 180 degrees. As the ASA congruency does not possess the necessity for knowing the angles, it is easier to use for proving the congruency of the triangles. The angle, side, the angle can be viewed as the formation of angles with the help of two lines and the same transversal. It can be dealt with with the help of algebra as it is associated with two pairs of angles that are congruent. ASA included parallel lines and geometric figures, only.

What is AAS?

When the vertices among two triangles contain correspondence in one-to-one such as two angles along with the opposite side of one of the angles in one triangle are congruent to the angles that are corresponding and the side that is not included in the second triangle. Under this circumstance, both the triangles are proved to be congruent to each other. Thus, it can be said that if both the pairs of angles that are corresponding and the opposite side of theirs are equal in two triangles, congruency can be established among both the triangles.

It is the same theorem as that of ASA except for the fact that its usage is done when all the sides of the triangle are congruent to the sides that are corresponding in the other triangle. The AAS congruency is associated with the formula, C=A-B. this congruency incorporated the value of all the angels ranging from 0 degrees to 360 degrees. For undergoing AAS congruency, one needs to know the lengths of the sides of the triangles that are involved in the proof for congruency. The formation of angles in the angle, angle, side cannot be viewed as it has the involvement of an angle that is included.


Main Differences Between ASA and AAS

  1. The abbreviation for ASA is Angle, Side, Angle. On the other hand, the abbreviation for AAS is Angle, Angle, Side.
  2. ASA is the proof of congruency associated with two triangles with equal sides among equal corresponding angles. Whereas, AAS is the proof of congruency associated with two angles and the opposite side of theirs being congruent to angles corresponding and non-included side of another triangle.
  3. The representation of ASA congruency involves a side but AAS does not involve a side in its congruency representation.
  4. ASA is proof aligning to congruency. On the other hand, AAS is proof aligning to similarities.
  5. ASA can be defined as the formation of angles by both lines involving non-included angles and the same transversal, while AAS can be defined as the formation of angles by both lines involving an included angle and the same transversal.

Conclusion

Thus, it can be concluded by saying that ASA and AAS congruency are distinctly different from each other in terms of different parameters. They mainly differ from each other due to their position of sides, angles, and their difference in usage in places. Angle, Side, Angle points towards the included side and any both angles. On the other hand, Angle, Angle, Side points towards the non-included side as well as both the angles that are corresponding. The ASA congruency can be proved with the application of geometry whereas AAS can use trigonometry to determine its congruency.

References

  1. https://www.sciencedirect.com/science/article/abs/pii/S0387760481800654
  2. https://academicjournals.org/journal/IJPS/article-abstract/66F5B4A12933
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