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Triangle Congruence Theorem
Triangle congruence theorem consists of five theorems that prove the congruence of two triangles. Two triangles are said to be congruent or the same if the shape and size of both the triangles are the same i.e. the corresponding sides placed in the same position and the corresponding angles placed in the same position of both triangles are the same. Let us learn more about the triangle congruence theorem, the different theorems, and solve a few examples.
What is Triangle Congruence Theorem?
Triangle congruence theorem or triangle congruence criteria help in proving if a triangle is congruent or not. The word congruent means exactly equal in shape and size no matter if we turn it, flip it or rotate it. In geometry, if the shapes are superimposed on each other, they are termed congruent figures, for example, triangles and quadrilaterals can be congruent.
The following are the triangle congruence theorems or the triangle congruence criteria that help to prove the congruence of triangles.
 SSS (Side, Side, Side)
 SAS (Side, Angle, Side)
 ASA (Angle, Side, Angle)
 AAS (Angle, Angle, Side)
 RHS (Right angleHypotenuseSide or the Hypotenuse Leg theorem)
Angle Side Angle Congruence Theorem
AngleSideAngle (ASA) theorem states that if two triangles are considered congruent, then the two corresponding angles equal to each other will include a corresponding side that is equal to each other. 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
Does this mean ΔABC and ΔDEF are congruent? Let us superimpose both the triangles and align them according to the angles and sides. 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). Hence, we notice that both the triangles are of the same shape and size, we can say that ΔABC ≅ ΔDEF.
Side Angle Side Congruence Theorem
The SideAngleSide (SAS) congruence theorem states that, if two corresponding sides of two triangles are equal including the corresponding angles formed by these sides then the two triangles are congruent. Let us see the proof of the theorem:
Given: AB=PQ, BC=QR, and ∠B=∠Q. To prove: ΔABC ≅ ΔPQR
Just like in ASA, let us superimpose triangles here again. Place the triangle ΔABC over the triangle ΔPQR such that B falls on Q and side AB falls along the side PQ. Since AB=PQ, so point A falls on point P, ∠B=∠Q, so the side BC will fall along the side QR, and BC=QR, so point C falls on point R. Thus, BC coincides with QR and AC coincides with PR. Therefore, we can say ΔABC ≅ ΔPQR.
Side Side Side Congruence Theorem
SideSideSide (SSS) congruence theorem states that if three sides of a triangle is equal to the corresponding sides of the other triangle, the two triangles are said to be congruent. Let us see the proof of the theorem:
Given: AB = DE, BC = EF, and AC = DF. To prove: ∆ABC ≅ ∆DEF.
We know that the three sides of both the triangles are of the same size and length. When we superimpose both the triangles, DE will be placed on AB, EF will be placed on BC, and DF will be placed on AC. Which makes it AB = DE, BC = EF, and AC = DF. Therefore, we can say that ∆ABC ≅ ∆DEF.
Angle Angle Side Congruence Theorem
AngleAngleSide congruence theorem states that if two angles of a triangle with a side that is not included between the angles is equal to the corresponding angles and side of the other triangle, they are considered to be congruent. Let us see the proof of the theorem:
Given: AB = DE, ∠B=∠E, and ∠C=∠F. To prove: ∆ABC ≅ ∆DEF
If both the triangles are superimposed on each other, we see that ∠B=∠E and ∠C=∠F. And the nonincluded sides AB and DE are equal in length. Therefore, we can say that ∆ABC ≅ ∆DEF.
RHS Congruence Theorem
RHS, Right angleHypotenuseSide or the Hypotenuse Leg theorem, states that if the hypotenuse and side of one rightangled triangle are equal to the hypotenuse and the corresponding side of another rightangled triangle, the two triangles are congruent. When we keep the hypotenuse and any one of the other 2 sides of two right triangles equal, we are automatically getting three similar sides, as all three sides in a right triangle are related to each other and that relation is popularly known as Pythagoras theorem. Let us see the proof of the theorem:
Given: Hypotenuse side and angles are equal. Tp prove: ∆ABC ≅ ∆PQR
If in any two triangles, the hypotenuse side doesn't measure the same or doesn't have a measurement we use (hypotenuse)^{2} = (base)^{2} + (perpendicular)^{2}. Both the triangles need to have a 90° right angle equal and the hypotenuse to be equal. In the above image, as we can see the measurements are the same for both triangles. Hence, we can say ∆ABC ≅ ∆PQR.
Related Topics
Listed below are a few topics related to the triangle congruence theorem, take a look:
Examples on Triangle Congruence Theorem

Example 1: Ben has four squares with the following side lengths: Square A, side = 7 inches, Square B, side = 9 inches, Square C, side = 9 inches, Square D, side = 8 inches. He wants two squares that can be placed exactly one over the other. Can you help him choose the congruent squares?
Solution: Squares with the same sides will superimpose on each other because they will be congruent. So, Ben should find two squares whose side lengths are exactly the same. In the given list, we can see that Square B and Square C have sides of the same length, that is, 9 inches. Therefore, Ben can choose Square B and C because they can be placed exactly one over the other.

Example 2: Identify the rule of congruence in the triangles ABC and DCB, and prove that they are congruent triangles.
Solution:
ln ΔABC and ΔDCB, let us try to identify any three parts of one triangle equal to the corresponding parts of the other triangle.
By observation, we find that AB = DC, AC = DB, and BC is common.
Thus by SSS rule of congruence, ΔABC ≅ ΔDCB.
FAQs on Triangle Congruence Theorem
What is Meant by Triangle Congruence Theorem?
Triangle congruence theorem or triangle congruence criteria help in proving if a triangle is congruent or not. There are 5 triangle congruence theorems  Side Side Side Theorem, Side Angle Side Theorem, Angle Side Angle Theorem, Angle Angle Side Theorem, and Right angleHypotenuseSide or the Hypotenuse Leg theorem.
What is the 4 Triangle Congruence Theorem?
The 4 triangle congruence theorems that help in finding if the triangles are congruent or not are:
 SSS (Side, Side, Side)
 SAS (Side, Angle, Side)
 ASA (Angle, Side, Angle)
 AAS (Angle, Angle, Side)
What is SSS, SAS, ASA, and AAS Triangle Congruence Theorem?
The 4 different triangle congruence theorems are:
 SSS: Where three sides of two triangles are equal to each other.
 SAS: Where two sides and an angle included in between the sides of two triangles are equal to each other.
 ASA: Where two angles along with a side included in between the angles of any two triangles are equal to each other.
 AAS: Where two angles of any two triangles along with a side that is not included in between the angles, are equal to each other.
What is the RHS Triangle Congruence Theorem?
RHS, Right angleHypotenuseSide or the Hypotenuse Leg theorem, states that if the hypotenuse and side of one rightangled triangle are equal to the hypotenuse and the corresponding side of another rightangled triangle, the two triangles are congruent.
How Do You Tell if a Triangle is ASA or AAS?
Both the triangle congruence theorems deal with angles and sides but the difference between the two is ASA deals with two angles with a side included in between the angles of any two triangles. Whereas AAS deals with two angles with a side that is not included in between the two angles of any two given triangles.
What are the Properties of Triangle Congruence Theorem?
There are 5 properties of congruent triangles which form the conditions to determine if they are congruent or not. They are:
 SSS Criterion for Congruence
 SAS Criterion for Congruence
 ASA Criterion for Congruence
 AAS Criterion for Congruence
 RHS Criterion for Congruence
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