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 with the congruency associated with an included side and any two angles. AAS aligns with the congruency associated with a non-included side and two corresponding angles.
Key Takeaways
- ASA (Angle-Side-Angle) and AAS (Angle-Angle-Side) are both methods for proving two triangles congruent.
- ASA requires two angles and the included side to be congruent, while AAS requires two angles and a non-included side.
- ASA and AAS provide valid proof for congruency, but the order of the elements is different.
ASA vs AAS
In ASA, two triangles are considered congruent if they have two corresponding angles, and the included side between those angles is congruent. In AAS, two triangles are considered congruent if they have two corresponding angles, and a non-included side between those angles is congruent.
In ASA, the requirement that the triangles are congruent is satisfied if the vertices of two triangles are in one-to-one correspondence, 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 triangle, respectively.
AAS or angle, angle, and side congruency are 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 similar angles. It involves two lines intersecting each other.
Comparison Table
| Parameters of comparison | ASA | AAS |
|---|---|---|
| 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 triangles contain an equal side incorporated among equal angles that correspond 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 triangles are congruent to each other. Both 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 congruent pairs of congruent angles.
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 correspond and the opposite side of theirs are equal in two triangles, congruency can be established among both 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 correspond in the other triangle.
The AAS congruency is associated with the formula C=A-B. This congruency incorporated the value of all the angles 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, and side cannot be viewed as it has the involvement of an angle that is included.
Main Differences Between ASA and AAS
- The abbreviation for ASA is Angle, Side, Angle. On the other hand, the abbreviation for AAS is Angle, Angle, Side.
- ASA is the proof of congruency associated with two triangles with equal sides among equal corresponding angles. At the same time, AAS is the proof of congruency associated with two angles and the opposite side of theirs being congruent to angles corresponding to and non-included side of another triangle.
- The representation of ASA congruency involves a side, but AAS does not involve a side in its congruency representation.
- ASA is proof of aligning to congruency. On the other hand, AAS is proof of aligning with similarities.
- 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.
- https://www.sciencedirect.com/science/article/abs/pii/S0387760481800654
- https://academicjournals.org/journal/IJPS/article-abstract/66F5B4A12933
Last Updated : 13 July, 2023
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Emma Smith holds an MA degree in English from Irvine Valley College. She has been a Journalist since 2002, writing articles on the English language, Sports, and Law. Read more about me on her bio page.
The comparison of the two methods was very well executed and informative. This serves as a great Beginners guide to understanding the same.
The inclusion of the triangles is an axiom is to a great extent warranted, considering the angles and sides. Although the importance of the congruency proofs is undeniable, the reliance on algebra seems unrealistic.
It seems to me that ASA and AAS are two sides of the same coin. Both can be used to obtain congruent triangles, however, they are different in terms of what and how many information is needed.
I fail to comprehend why ASA involves the inclusion of sides while AAS does not. This seems like the only difference between the two methods.
It’s interesting to see that ASA and AAS are two of the methods for proving congruency among triangles. These methods are truly fascinating and it’s great to get a deeper understanding of how they differ and where they overlap.
The inclusion of the side in ASA proves to be an entirely different concept to AAS despite the similarities – it’s a wonder they’re both so linked! Amazing to learn about.
This is incredible, what an important difference. It’s truly fascinating to learn about these triangles and their similarities and differences.