Congruence of Triangles: SSS, SAS, ASA & RHS Rules

Congruence of Triangles states when every corresponding side and interior angle of a triangle are congruent, it is said to be in congruence (of the same length). In simple terms, when one object is placed on top of another, it appears to be the same figure or copies of each other. Then the triangles are said to be congruent.

Key Terms: Congruence of Triangles, Triangles, Properties of Congruence of Triangles, Congruent Triangles, Congruence of Triangles Rules

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Key Highlights

• The congruence of triangles depends on the measurements of their angles and sides.
• The sides of the triangles are used to determine the size , and the angles of a triangle are used to determine the shape of the figure.
• For two triangles to be congruent, all three sides and angles, respectively, should be congruent. 
• The rules of congruency are divided into six categories: SSS (Side-Side-Side), SAS (Side-Angle-Side), ASA (Angle-Side-Angle), AAS (Angle-Angle-Side), and RHS (Right angle-hypotenuse-side).
•  CPCT, which stands for “Corresponding Parts of Congruent Triangles”, is used to describe congruent triangles.

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Congruent Triangles

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Two triangles are said to be congruent if all the corresponding sides and angles of the triangles are equal in measure. Every triangle is made up of six measurements: three sides and three angles.

• The congruency of the triangles can be determined depending on the values of these measurements. 
• Each side and angle is of the same shape and size.
• Congruent Triangles superimpose angle to angle and side to side.
• It is denoted by the symbol “≅”.
• The real-life application of congruent triangles includes carpet designs, stepping stone patterns, and architectural designs.
• CPCT (Corresponding Parts of Congruent Triangles) proves the congruency of triangles that overlap with each other completely.

Congruent Triangles

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Congruence of Triangles Rules

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The following are the rules of congruence triangles that determines whether the triangles are congruent or not. The property of congruency is based on the number of sides and angles that make up a congruent pair. 

SAS (Side-Angle-Side)

If two sides and the involved interior angle of one triangle are equivalent to the sides and the involved angle of the other triangle, the triangle declares congruence with each other and follows the SAS (Side-Angle-Side) rule.

In the below-given figure, we can see that, 

= AB = DE

= BC = EF

= ∠B = ∠C. 

Hence, Δ ABC ≅ Δ DEF

SAS (Side-Angle-Side)

ASA (Angle Side Angle)

Two triangles are congruent if two angles and the involved side of one triangle are equivalent to angles and the side of the other triangle. Then, the two triangles will satisfy the ASA (Angle Side Angle) congruency. 

In the below-given figure, we can see that,
= BC = EF
= ∠B = ∠E
= ∠C = ∠F
Hence,  ΔACB ≅  ΔDEF.

ASA (Angle Side Angle)

SSS (Side Side Side)

Two triangles are said to be congruent if all three sides of one triangle are equal to all three sides of the other triangle. Then, the two triangles will satisfy the SSS (Side Side Side) congruency. 

In the below-given figure, we can see that,
= AB = PQ
= BC = QR
= AC = PR
Hence,  ΔACB ≅  ΔPQR.

SSS (Side Side Side)

AAS (Angle Angle Side)

Two triangles are congruent if one pair of corresponding sides and either of the two pairs of angles is equivalent to each other. Then, the two triangles will satisfy the AAS (Angle Angle Side) congruency. 

• The angle sum property of triangles suggests that the sum of three angles in a triangle is 180°, which is the criterion for this principle.
• As a result, if two triangles have the same measure, the third side is automatically equal, resulting in ideally congruent triangles.
• From the below figure, we can say that, 

= AC = PR
= ∠C = ∠R
= ∠B = ∠Q
Hence, ΔACB ≅  ΔPQR

AAS (Angle Angle Side)

RHS (Right Angle Hypotenuse)

If the hypotenuse and one side of one right-angled triangle measure the same as the hypotenuse and one side of the other right-angled triangle, then the pair of two triangles is congruent with each other.

From the below-given figure of a right-angled triangle, we get 
∠B = ∠Q (right angles)
CA = RP and 
AB = PQ. 
Hence, ΔACB ≅ ΔPRQ

RHS (Right Angle Hypotenuse) 

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Properties of Congruent Triangles

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Various properties of Congruent Triangles are as follows:

• Congruent Triangles overlap with each other and form mirror images of each other.
• These triangles are arranged in the proper orientation.
• In these types of triangles, corresponding parts of congruent triangles are equal and have the same area.

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Limitations of Congruence of Triangles

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The following combination of the parameters does not result in the congruence of triangles.

• According to this rule, if all of a triangle's corresponding angles measure equal, the triangles will be roughly the same shape, but not necessarily the same size.
• It will be a case of two identical triangles, one of which is larger than the other.
• Given two sides and a non-involved angle, two different triangles will likely emerge, each convincing the values but insufficient to demonstrate congruence.