ASA Congruence Rule

ASA congruence rule states that if two angles of one triangle, and the side contained between these two angles, are respectively equal to two angles of another triangle and the side contained between them, then the two triangles are considered to be congruent. For two triangles to be congruent, the major elements, mainly three angles, and the three sides of a triangle should be equal to the corresponding angles and the corresponding sides of another triangle. Thus, we can say that the two triangles are congruent if they are of the same size and same shape. It is not necessary that all the six corresponding elements of both the triangles must be equal in order to determine their congruency. The 5 congruence rules include SSS, SAS, ASA, AAS, and RHS.

In this lesson, let's learn about the ASA congruence rule in detail.

ASA Congruence rule stands for Angle-Side-Angle. Under this rule, two triangles are said to be congruent if any two angles and the side included between them of one triangle are equal to the corresponding angles and the included side of the other triangle. Look at the image given below to determine if the two given triangles, Δ ABC and ΔXYZ are congruent by the ASA rule.

Under ASA criterion, Δ ABC ≅ ΔXYZ, as ∠B = ∠Y, ∠C = ∠Z, and the side BC = YZ. Since Δ ABC ≅ ΔXYZ, then the third angle ∠A and the other two sides of Δ ABC are bound to be equal to the corresponding angle ∠X and the sides of ΔXYZ.

Consider the two triangles, ABC and DEF in which ∠ B = ∠ E, ∠ C = ∠ F, and BC = EF

To Prove: Δ ABC ≅ Δ DEF.

Proof:

For proving the congruence of the two triangles three cases arise.

Case (i):

AB = DE (Assumed), ∠ B = ∠ E (Given), BC = EF (Given). Thus, Δ ABC ≅ Δ DEF (by SAS)

Case (ii):

Let if possible AB > DE and thus, we can take a point P on AB such that PB = DE. Now consider Δ PBC ≅ Δ DEF.

In Δ PBC and Δ DEF, PB = DE (By construction), ∠ B = ∠ E (Given), BC = EF (Given).

Thus, Δ PBC ≅ Δ DEF, by the SAS congruence rule.

Since the triangles are congruent, their corresponding parts will be equal. Thus, ∠ PCB = ∠ DFE. But, we are given that ∠ ACB = ∠ DFE. So, ∠ ACB = ∠ PCB. This is possible only if P coincides with A. or, BA = ED So, Δ ABC ≅ Δ DEF (by SAS axiom)

Case (iii):

If AB < DE, we can choose a point M on DE such that ME = AB and repeating the arguments as given in Case (ii), we can conclude that AB = DE and so, Δ ABC ≅ Δ DEF.

Suppose, now in two triangles two pairs of angles and one pair of corresponding sides are equal but the side is not included between the corresponding equal pairs of angles. Are the triangles still congruent? You will observe that they are congruent. We know that the sum of the three angles of a triangle is 180° and thus if two pairs of angles are equal, the third pair(180° – sum of equal angles) is also equal. Therefore, we can conclude that the two triangles are congruent if any two pairs of angles and one pair of corresponding sides are equal. We may even call it the AAS Congruence Rule.

In order to check if the given triangles are congruent, the best way is to superimpose them, put them one over the other. To apply the ASA congruence rule for any two given triangles, follow the steps as given below:

• Step 1: Look at the triangles and check for the given angles and sides.
• Step 2: Make comparisons if two angles with one included side of a triangle are equal to the corresponding two angles and included side of the other triangle.
• Step 3: If the above conditions are satisfied, the given triangles are considered congruent by the ASA rule.

Let's look at the examples given in the examples section for better clarity.

Important notes

• Two triangles are said to be congruent if the six elements (three sides and three angles) of one triangle are equal to the corresponding six elements of the other triangle.
• There are overall five conditions or rules to determine if two triangles are congruent which include SSS, SAS, ASA, AAS, and RHS.
• The congruency is expressed using the symbol (≅).
• In case the two triangles are congruent then their perimeters and areas are also equal.

Related Topics

Also, check topics related to as congruence rule:

• Similarity in Triangles
• Transitive Property of Congruence
• What is Congruence?
• Corresponding Parts of Congruent Triangles

Give the Statement of ASA Congruence Rule.

The statement of ASA congruence rule is given as: "If two angles and the included side of one triangle are equal to two angles and included side of another triangle, then the triangles are congruent".

What Is AAS and ASA Congruence Rule?

ASA and AAS are two postulates that determine if two triangles are congruent. ASA stands for “Angle, Side, Angle”, that is, any two angles and the included side while AAS means “Angle, Angle, Side”, that is, the two corresponding angles and the non-included side

How Can You Identify the ASA Congruence in Triangles?

We can identify if the two triangles are congruent by checking if the parts of one triangle are equal to the corresponding parts of the other triangle. As the name says, ASA can be identified in case the two angles and the included side of one triangle are equal to two angles and included side of another triangle.

How to Prove Congruence of Triangles using ASA Congruence Rule?

• Step 1: Observe the two given triangles for their angles and sides.
• Step 2: Compare if two angles with one included side of a triangle are equal to the corresponding two angles and included side of the other triangle.
• Step 3: The given triangles are considered congruent by the ASA rule if the above conditions get satisfied.