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How to Prove Triangles Congruent - SSS, SAS, ASA, AAS Rules




In this lesson, we will learn

  • the SSS, SAS, ASA and AAS rules
  • how to use two-column proofs to prove triangles congruent

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Congruent Triangles

Congruent triangles are triangles that have the same size and shape. This means that the corresponding sides are equal and the corresponding angles are equal.

We can tell whether two triangles are congruent without testing all the sides and all the angles of the two triangles. In this lesson, we will consider the four rules to prove triangle congruence. They are called the SSS rule, SAS rule, ASA rule and AAS rule. In another lesson, we will consider a proof used for right triangles called the Hypotenuse Leg rule. As long as one of the rules is true, it is sufficient to prove that the two triangles are congruent.

Side-Side-Side (SSS) Rule

Side-Side-Side is a rule used to prove whether a given set of triangles are congruent. 

The SSS rule states that 

If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.

In the diagrams below, if AB = RP, BC = PQ and CA = QR, then triangle ABC is congruent to triangle RPQ.

sss rule



Side-Angle-Side (SAS) Rule

Side-Angle-Side is a rule used to prove whether a given set of triangles are congruent. 

The SAS rule states that 

If two sides and the included angle of one triangle are equal to two sides and included angle of another triangle, then the triangles are congruent.

An included angle is an angle formed by two given sides.

included and non-included angle
Included Angle               Non-included angle

For the two triangles below, if AC = PQ, BC = PR and angle C = angle P , then using the SAS rule, triangle ABC is congruent to triangle QRP

congruent triangles



Angle-Side-Angle (ASA) Rule

Angle-side-angle is a rule used to prove whether a given set of triangles are congruent. 

The ASA rule states that 

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.

Angle-Angle-Side (AAS) Rule

Angle-angle-side is a rule used to prove whether a given set of triangles are congruent. 

The AAS rule states that 

If two angles and a non-included side of one triangle are equal to two angles and a non-included side of another triangle, then the triangles are congruent.

In the diagrams below, if AC = QP, angle A = angle Q, and angle B = angle R, then triangle ABC is congruent to triangle QRP.

The following video will explain three ways to prove triangles congruent - A lesson on SAS, ASA and SSS


 



Using Two Column Proofs to Prove Triangles Congruent

Triangle Congruence by SSS - How to Prove Triangles Congruent
Side Side Side Postulate
If three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent.

Triangle Congruence by SAS - How to Prove Triangles Congruent
SAS Postulate
If two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle, then the two triangles are congruent.

Prove Triangle Congruence with ASA Postulate
Angle Side Angle Postulate
It two angles and the included side of one triangle are congruent to two angles and the included side of another triangle, then the two triangles are congruent.

Triangle Congruence by AAS Postulate

Angle Angle Side Postulate
It two angles and a non-included side of one triangle are congruent to two angles and a non-included side of another triangle, then the two triangles are congruent.


 



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