Criteria For Congruence Of Triangles: SAS, ASA, SSS, RHS, and Other

In this module, we will discuss the Criteria for the Congruence of Triangles.

We make use of triangular structures and patterns frequently in day-to-day life. So, it is rewarding to find out when two triangular shapes will be congruent. We know that 2 triangles are congruent if their corresponding sides and angles are equal.

Criteria For Congruence Of Triangles

SAS (Side-Angle-Side) Congruence Rule:

Let us now learn the first criterion of congruency, the SAS congruence rule.

Two triangles are congruent if two sides and the included angle of one triangle is equal to the sides and the included angle of the other triangle.

In triangles XYZ and ABC, which are congruent by the SAS criterion?

Side XY is equal to side AB, side YZ is equal to side BC, and included angles Y and B are equal.

ASA (Angles Side Angle) criterion:

Now let us discuss the ASA (Angles Side Angle) criterion of congruency.

If under a correspondence, two angles and the included side of a triangle are equal to two corresponding angles and the included side of another triangle, then the triangles are congruent.

Let us take an example.

If in two triangles, PQR and XYZ, angle Q is equal to angle Y

Angle R is equal to angle Z and QR is equal to YZ.

Then these two triangles are congruent by the ASA criterion. As in the two triangles, two corresponding angles and the included side is equal.

SSS (Side-Side-Side) criterion:

Let us now discuss the SSS (Side-Side-Side) criterion of congruency.

If under a given correspondence, the three sides of one triangle are equal to the three corresponding sides of another triangle, then the triangles are congruent.

If in two triangles ABC and PQR, AB is equal to PQ, BC is equal to QR and CA is equal to RP, then we can say two triangles ABC and PQR are congruent by SSS (Side-Side-Side) congruency.

Congruence Among Right-angled Triangles

Let us now discuss Congruence among Right-Angled Triangles.

Congruence in the case of two right triangles deserves special attention.

In such triangles, the right angles are equal.

So, the congruence criterion becomes easy.

This congruence leads to the RHS Congruence criterion:

If under a correspondence, the hypotenuse and one side of a right-angled triangle are respectively equal to the hypotenuse and one side of another right-angled triangle, then the triangles are congruent.

In triangles, ABC and PQR, angles B and Q are right angles. And AC is equal to PR and AB is equal to PQ. So, we can say triangles ABC and PQR are congruent by RHS congruency.

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Frequently Asked Questions (FAQs)

What does the SAS criterion for triangle congruence state? How is the ASA criterion used to determine triangle congruence? What is the SSS criterion for triangle congruence?