We know that triangles can be congruent and also similar to each other. And we always consider all its sides and angles to prove it. But not anymore, here we will learn a triangle criterion by which we can easily prove congruent triangles.
In this section, we will take a look at ASA Theorem and understand how to prove congruence and similarity between triangles without all sides and angles of triangles.
ASA Theorem geometry
In geometry, two triangles are congruent when either all the sides of one triangle are equal to all sides of another triangle respectively. Or all the three angles of both the triangles should be equal respectively. But with the ASA criterion, we can show congruent triangles with the help of two angles and one side of each triangle.
ASA theorem, as the name suggests, considers two angles and one side of one triangle equal to another triangle respectively. Here two adjacent angles and the included side between these angles are taken. But one should remember that ASA is not the same as AAS. As ASA has the included side of the two triangles, but in AAS the selected side is the unincluded side of both angles.
ASA triangles, StudySmarter Originals
ASA similarity and congruence theorem
We can easily find similar triangles and congruent triangles with the help of the ASA similarity and congruence theorem.
ASA similarity theorem
We know that if two triangles are similar then all the corresponding sides are in proportionality and all the corresponding pairs are congruent from the definition of similar triangles. However, in order to ensure the similarity of two triangles we only need information about two angles with the ASA similarity theorem.
ASA similarity theorem : Two triangles are similar if two corresponding angles of one triangle are congruent to the two corresponding angles of another triangle. Also, the corresponding sides are proportional.
Mathematically we represent as, if 
then 
And
ASA Similarity triangles, StudySmarter Originals
Generally ASA similarity is more well known as the AA similarity theorem, as there is nothing further to check because of only one ratio of sides. Also when two angle measures are given, we can easily find the third angle as the total angle measure is
. So we can easily check the equality of corresponding angles of two triangles and determine the similarity of both triangles.
ASA congruence theorem
ASA congruence theorem stands for Angle-Side-Angle and gives the congruent relation between two triangles.
ASA congruence theorem: Two triangles are congruent if two adjacent angles and the included side on one triangle are congruent to the two angles and included side of another triangle.
Mathematically we say that, if
then 
As the angles and sides are congruent they will also be equal. So
then
ASA congruence triangles, StudySmarter Originals
ASA theorem proof
Now let us take a look at ASA theorem proof for similarity and congruence.
ASA similarity theorem proof
For two triangles 
and 
it is given from the statement of ASA similarity theorem that 
Since 
then the corresponding parts of congruent triangles are congruent.
Also, it is given that
From equation (1) and equation (2) 
Since 
and 
forms corresponding angles and XY works as transversal 
Using Basic Proportionality Theorem in 
Basic Proportionality theorem states that if a line is parallel to one side of a triangle and it intersects the other two sides at two different points then those sides are in proportion.
From the construction of PQ we replace 
and 
in the above equation.
Similarly, 
Hence 
ASA congruence theorem proof
We are given from the statement of ASA congruence theorem that 
and 
ASA congruent triangles, StudySmarter Originals
To prove: 
To prove the above statement we will consider different cases.
Case 1: Assume that 
ASA congruent triangles with AB and LM equal, StudySmarter Originals
Case 2: Suppose 
Then we construct point X on AB such that 
ASA congruent triangle with constructed point X, StudySmarter Originals
We have 
and 
Using SAS congruence theorem 
Now it is given that 
And from the above congruence,
we get that 
So from equations (1) and (2), we get
Case 3: Suppose 
Then construct a point Y on LM such that 
and we repeat the same argument as in case 2.
ASA congruent triangles with constructed point Y, StudySmarter Originals
Hence we get that 
ASA Theorem example
Let us see some examples related to ASA theorems.
Calculate BD and CE in the given figure, if 
Solution:
In 
and 
as 
then 
because they are alternate interior angles. Also 
forms vertically opposite angles.
Then by ASA similarity theorem 
We also get from the ASA similarity theorem that 
Then by substituting all the given values in the above equation,
And 
Hence 
Calculate the value of x when 
Solution:
From the figure we can see that 
Then by ASA congruence theorem we get that 
Now substituting all the given values we get,
ASA Theorem - Key takeaways
- ASA congruence theorem: Two triangles are congruent if two adjacent angles and the included side on one triangle are congruent to the two angles and included side of another triangle.
- ASA similarity theorem: Two triangles are similar if two corresponding angles of one triangle are congruent to the two corresponding angles of another triangle. Also, the corresponding sides are proportional.
- ASA similarity is mostly known as the AA similarity theorem.
- ASA theorem is not the same as the AAS theorem.
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And 
and 
are already given in 
we can easily find 
by taking 
And the same will be the case for the triangle
such that 
and 
Also it is given that 
Then by using SAS congruence theorem we get that 
and 
from our assumption. And it is given that 
and 
So by SAS congruence theorem 
and also by looking at the figure this is not possible. So 
can only occur when both the points A and X coincides and 
and 
are equal. Hence we can consider only one triangle 
such that 
So this is the same as case 1, and from that we get that 
