Difference Between ASA and AAS (With Table)

Congruency is a condition when two figures are equal in shape and measurement. Triangles are congruent when they have equal side and angle. There are various condition to prove two triangles congruent. The conditions are: SSS, SAS, ASA, AAS and RHS.

ASA and AAS are two different condition which can be used to prove triangles congruent. The purpose of the condition are same but the method of proving is very different from each other.

ASA vs AAS

The main difference between ASA and AAS is the different method used to prove two triangles congruent. ASA is angle-side-angle and AAS is angle-angle-side.

ASA is a condition used to prove two triangles congruent. Full form of ASA is angle-side-angle. ASA is a easy way of proving congruency. The range of angle under this condition is 0° to 180°. This method cannot be used to prove similarity. Under this condition of congruency, any two angle inside the triangles and a corresponding side of two triangles should be equal. This condition can only be used in geometry and not trigonometry.

AAS is a condition used to prove two triangles congruent. Full form of AAS is angle-angle-side. AAS is comparatively difficult way of proving congruency. The range of angle under this condition is 0° to 360°. This condition is also used to prove similarity. This condition is more broader when compared with others. Under this condition of congruency, two triangles should have two angles and any side of triangle equal. It is not necessary to include side between the angles.

Comparison Table Between ASA and AAS

Parameters of ComparisonASAAAS
Full-formAngle-Side-AngleAngle-Angle-Side
DefinitionIt is defined as angles between two side of triangleIt is defined as any two angles and any other side of triangle
SideSide is between the angles are includedAny side can be included
DifficultyEasy to proveComparatively difficult
Used forProving congruency but not similarityProving congruency and similarity
Range of angles0° to 180°0° to 360°
ExampleFor proving congruency in triangle LMN and OPQ <L = <O , LM=OP and <M=<PTriangles LMN and OPQ are congruent if <L=<O , <N=<Q and LM = OP

What is ASA?

ASA is a condition of congruency. Full form of ASA is angle-side-angle. Under this condition of congruency two triangles should have two angles and corresponding side between angles equal. Formula of ASA is A=B-C. This condition is easier than other conditions.

ASA is a narrower condition. It cannot be used in proving similarity. ASA can only be used in geometry and not in trigonometry. ASA can be used to determine degree of similarity but not to prove similarity. Range of angle under this condition is 0° to 180°. For using this condition to prove two triangles congruent, it is important to know measurement of side and degree of angle.

For example: Two triangles ABC and PQR are said to be congruent under the condition ASA is when corresponding angles and a side between the angles in both the triangles are equal i.e <A=<P , AB=PQ and <B=<Q.

What is AAS?

AAS is a condition used to prove congruency and similarity of a triangle. AAS stands for angle-angle-side. Under this condition two triangles are said to be congruent if corresponding angles and any corresponding side are equal. It is not necessary that side between the angles are included in this condition. Formula of AAS is C=A-B This condition is comparatively difficult that other conditions.

AAS is a broader condition. It can be used to prove triangles similar as well. AAS can also be used in trigonometry along with geometry. Range of angle under AAS is 0° to 360°. For using this condition to prove two triangles congruent, it is important to know degree of angles of a triangle.

For example: Two triangles ABC and PQR are said to be congruent under the condition AAS is when corresponding angle and any one side in the both the triangles are equal i.e <A=<P , <B=<Q and AB=PQ/AC=PR.

Two triangles ABC and PQR are said to be similar under the condition AA is when any two angles of the triangle is equal.

Main differences between ASA and AAS

  1. ASA means angle-side-angle. While AAS means angle-angle-side.
  2. ASA is the condition where the side taken must be between the two angles taken. AAS is the condition where the side taken does not need to be between angles taken.
  3. ASA is a easier way to prove two triangles congruent. While AAS is comparatively difficult way.
  4. ASA includes range of angle 0° to 180°. While AAS includes angle of range 0° to 360°.
  5. ASA cannot be used to prove similarity. While AAS is used to prove similarity.

Conclusion

The conclusion can be drawn that triangles can be proved congruent by both the condition i.e ASA and AAS. Both are the tool for proving triangles congruent. Despite of having the same purpose both ASA and AAS are different from each other.

The method of proving, steps involved everything is different. In ASA side between the two angles taken are included. While in AAS any side of the triangle can be taken along with any two angle inside the triangle. In AAS it is not necessary to include side between the angle.

However, ASA is considered more simpler and easier when compared to AAS. AAS is also used to prove triangles similar by comparing corresponding angles of two triangles. ASA can not be used for proving similarity between triangles.

References

  1. https://rusmp.rice.edu/sites/g/files/bxs3761/files/inline-files/SAS%20and%20SSA%20conditions%20for%20congruent%20triangles.pdf
  2. https://ugei-ojs-shsu.tdl.org/ugei/index.php/ugei/article/view/12