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# ASA Theorem

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. Sothen

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

To prove: And

ASA triangle with constructed line , StudySmarter Originals

Now as two angles and are already given in we can easily find by taking And the same will be the case for the triangle

We will construct a line PQ in triangle such that and Also it is given that Then by using SAS congruence theorem we get 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

In and from our assumption. And it is given that and So by SAS congruence theorem

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

But from our assumption of and also by looking at the figure this is not possible. So can only occur when both the points A and X coincides and

So we are again left with the fact that 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

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.

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 theorem is used to find congruence between two triangles if two angles and included side are given.

If two angles and the included side are given and they are equal to the corresponding angles and side, then that triangles are ASA.

ASA theorem is proved with the help of the SAS theorem.

## Final ASA Theorem Quiz

Question

What is ASA 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.

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Question

Is ASA theorem same as AAS theorem?

No

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Question

If two corresponding pairs of angles of two triangles are given and both are congruent, the third pair of angles can be easily found.

True

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Question

Write 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.

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Question

ASA similarity theorem is mostly called as?

AA similarity theorem

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Question

Is the given figure congruent using ASA congruence theorem?

(flashcards) en-mathematics-pure maths-geometry-asa triangle common side

No

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Question

Which theorem can be used to prove the below two triangles congruent?

(flashcards) en-mathematics-pure maths-geometry-asa triangle common vertex

ASA congruence theorem

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Question

Can the ASA theorem be used to prove congruent between given two triangles?

(flashcards) en-mathematics-pure maths-geometry-congruent triangles not asa

Yes

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Question

Which side will be considered as an included side for the ASA theorem?

RS

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Question

Are the given two triangles congruent with ASA congruence theorem?

(flashcards) en-mathematics-pure maths-geometry-right angle triangle

Yes

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Question

Can similarity between given two triangles be found with the ASA similarity theorem?

(flashcards) en-mathematics-pure maths-geometry-triangle two congruent angles

Information missing

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