ASA Congruence Rule | Definition, Proof & Examples

ASA Congruence Rule: ASA stands for Angle-Side-Angle. It is one of the congruence tests used to test the congruence of two triangles. Other than ASA there are 4 more congruence rules i.e., SSS, SAS, AAS, and RHS.

Condition of Congruency of Two Triangles: Two triangles are said to be congruent if two angles and the included side of one triangle are equal to two angles and the included side of another triangle.

In this article, we will learn about the ASA Congruence Rule including its proof, applications, and examples related to it.

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What is Congruence?

Congruence is a term used in mathematics to describe a relationship between geometric figures or mathematical objects that have the same shape and size. In other words, if one figure can be transformed into the other by a combination of translations, rotations, and reflections without changing its shape or size, then they are considered congruent.

For example, two triangles are congruent if their corresponding sides are equal in length and their corresponding angles are equal in measure. Congruence is often denoted by the symbol "≅".

Read More: Congruence of Triangles

What is ASA Congruence Rule?

According to the ASA Congruence Rule, if two angles and the included side of one triangle are equal in measure to two angles and the included side of another triangle, then the two triangles are congruent.

To apply the ASA Congruence Rule, you must have the following conditions met:

• Two angles in one triangle are congruent (equal in measure) to two angles in another triangle.
• The side that is between the two angles in one triangle is congruent to the side between the corresponding angles in the other triangle.

ASA Congruence Rule Definition

The Angle-Side-Angle (ASA) Congruence Rule is a criterion in geometry that determines the congruence of two triangles. According to this rule, if two angles and the included side of one triangle are exactly equal to two angles and the included side of another triangle, then the two triangles are congruent

Criteria for ASA Congruence Rule

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Under ASA criterion of congruence rule, ∆ABC ≅ ∆XYZ if

• ∠B = ∠Y,
• ∠C = ∠Z,
• BC = YZ

By CPCT property [Corresponding Parts of Congruent Triangles are equal], we can further imply the following as well:

• ∠A = ∠X
• AB = XY
• AC = XZ

Proof of ASA Congruence Rule

From the given two triangles, ABC and DEF, in which:

∠B = ∠E, and ∠C = ∠F and the BC = EF

To Prove: ∆ ABC ≅ ∆ DEF

To prove the congruence of the two triangles, the three cases involved are

• Case 1: AB = DE
• Case 2: AB > DE
• Case 3: AB < DE

Case 1: Let AB = DE

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• AB = DE (Assumed)
• ∠B = ∠E [Given]
• BC = EF [Given]

So, from SAS Rule we get, ∆ ABC ≅ ∆ DEF

Case 2: AB > DE.

Now take a point P on AB such that it becomes PB = DE.

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Now consider ∆ PBC and ∆ DEF,

It is noted that in triangle PBC and triangle DEF,

From construction, PB = DE

Given,∠ B = ∠ E

BC = EF

So, we conclude that, from the SAS congruence axiom

∆ PBC ≅ ∆ DEF

Since the triangles are congruent, their corresponding parts of the triangles are also equal.

So, ∠PCB = ∠DFE

But, we are provided with that

∠ACB = ∠DFE

So, we can say ∠ACB = ∠PCB

This condition is possible only if P coincides with A or when BA = ED

So, ∆ ABC ≅ ∆ DEF (From SAS axiom)

Case 3: If AB < DE,

Let's take a point M on DE such that it becomes ME = AB,

Repeating the arguments as given in Case (ii), we can conclude that AB = DE and so we get

∆ ABC ≅ ∆ DEF.

ASA and AAS Congruence Rule

The key difference between ASA and AAS congruence rule is listed as follows:

Criteria	ASA Congruence Rule	AAS Congruence Rule
Components	Two angles and sides between them	Any two angles and any one side.
Sequence	The congruent elements must follow an angle-side-angle sequence in both triangles.	Sequence of angle and sides doesn't matter.
Example	Consider two right triangles, △ABC (right angle at B) and △DEF (right angle at E). Then, ∠A = ∠D, ∠B = ∠E AB = DE ⇒ △ABC ≅ △DEF (by ASA)	Consider two triangles, △ABC and △DEF. Then, ∠A = ∠D ∠C = ∠F AC = DF ⇒ △ABC ≅ △DEF (by AAS)

Note: If two angles of a triangle are equal, then, by the angle sum property of a triangle, we can easily conclude that the third angle is also equal. Thus, if any triangle is proven to be congruent by the ASA criterion, it can also be easily proved by the AAS criterion.

People Also Read:

• RHS Congruence Rule
• SSS Congruence Rule
• SAS Congruence Rule

ASA Congruence Rule Class 9

In Class 9 geometry, the Angle-Side-Angle (ASA) Congruence Rule is a very important principle used to establish the congruence of two triangles. This rule specifies that if two angles and the included side of one triangle are equal to two angles and the included side of another triangle, then the two triangles are congruent.

• Geometric Proofs: The ASA rule is crucial for proving the congruence of triangles in geometric proofs. It is a direct method to demonstrate that two triangles are identical in shape and size, based on their angles and a side.
• Problem-Solving: Students learn to apply the ASA rule in various geometric problems, including those that require finding missing angles or sides, or proving certain properties about geometric figures.
• Theoretical Foundation: Understanding ASA congruence helps students grasp the concept of triangle congruence more broadly, preparing them for more complex geometry and trigonometry topics.

ASA Congruence Rule Solved Examples

Problem 1: Prove that △CBD ≅ △ABD from the given figure.

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Solution:

Given: AD∥EC, BD≅BC

To prove: △ABD ≅ △EBC

Proof:

AD ⊥ EC [Given]

∠D = ∠C [Alternate Interior angles theorem]

BD = BC [Given]

∠ABD = ∠EBC [Vertical angles congruence theorem]

Thus, by ASA Congruence criteria,

△ABD ≅ △EBC

ASA Congruence Rule: Practice Problems

Q1: In the diagram, AB ⊥ AD, DE ⊥ AD, and AC ≅ DC. Prove that △ABC ≅ △DEC.

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Q2: Use the ASA congruence theorem to prove that △NQM ≅ △MPL.

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