Angle Side Angle

Angle-Side-Angle is also called ASA criterion which means if two triangles are congruent 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. Angle side angle is one of the conditions for two triangles to be congruent. The other conditions are SSS, SAS, AAS, and RHS. In this section, we will explore the ASA rule, the formula, and the congruence theorem using real-life examples.

By definition, an angle side angle states that if two angles of one triangle, and the side between these two angles, are respectively equal to the two angles and the side between the angles of another triangle, then the two triangles will be congruent to each other by ASA rule. Two triangles are said to be congruent when:

• Three sides of one triangle will be (respectively) equal to the three sides of the other.
• Three angles of one triangle will be (respectively) equal to the three angles of the other.

However, in order to be sure that two triangles are congruent, we do not necessarily need to have information about all sides and all angles. Let us understand ASA with an example. Consider the following two triangles, Δ ABC and Δ DEF:

We are given that,

BC = EF
∠B = ∠E
∠C = ∠F

We say that by ASA criterion: Δ ABC ≅ ΔDEF.

ASA Congruence rule states that 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.

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

Case 1: AB = DE (Assumed), ∠ B = ∠ E (Given), BC = EF (Given). Thus, Δ ABC ≅ Δ DEF (by SAS)

Case 2: 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 3: If AB < DE, we can choose a point M on DE such that ME = AB and repeating the arguments as given in Case 2, 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° – the 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.

Angle side angle theorem states that two triangles are congruent if two angles and the included side of one triangle are equal to two angles and the included side of the other triangle. Let us see the proof of the ASA theorem:

Consider the following two triangles, Δ ABC and Δ DEF.

We are given that,

BC = EF
∠B = ∠E
∠C = ∠F

Can we say that ΔABC and ΔDEF are congruent?

Let us first do a thought experiment and try to superimpose ΔDEF on ΔABC. Align EF exactly with BC. Since ∠B = ∠E, the direction of ED will be the same as the direction of BA. Similarly, since∠C = ∠F, the direction of FD will be the same as the direction of CA. This means that the point of intersection of ED and FD (which is D) will coincide exactly with the point of intersection of BA and CA (which is A). Thus, since all the three vertices of the two triangles (can be made to) respectively coincide, the two triangles are congruent by angle side angle triangle congruence theorem. ΔABC ≅ ΔDEF

ASA formula is one of the criteria used to determine congruence. ASA congruence criterion 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 will be congruent.

Related Topics

Listed below are a few topics related to angle side angle, take a look.

• What is Congruence?
• Corresponding Parts of Congruent Triangles

What is Angle Side Angle?

Angle side angle also knows as ASA Criterion means if two triangles are congruent 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.

How Can You Identify the Angle Side Angle 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 Do You Find the Angle Angle Side?

In angle-angle side(AAS) if two angles and the one non-included side of one triangle are congruent to two angles and the non-included side of another triangle, then these two triangles are congruent.

What is the Angle Side Angle Theorem?

ASA congruence criterion 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 will be congruent.

Give the Statement of Angle Side Angle Congruence Rule.

The statement of the angle side angle 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".

How to Prove Congruence of Triangles using ASA Congruence Rule?

To prove the given triangles to be congruent using the ASA congruence rule by the steps given below:

• 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.