How to prove two triangles are congruent

Q: What are the 4 ways to prove triangles are congruent?

These four criteria used to test triangle congruence include: Side – Side – Side (SSS), Side – Angle – Side (SAS), Angle – Side – Angle (ASA), and Angle – Angle – Side (AAS). There are more ways to prove the congruency of triangles, but in this lesson, we will restrict ourselves to these postulates only.

Q: What are 5 ways to prove a triangle is congruent?

There are 5 main rules of congruency for triangles:.

Q: How do you know △ ABC ≅ △ DEF explain?

If two angle in one triangle are congruent to two angles of a second triangle, and also if the included sides are congruent, then the triangles are congruent. Using labels: If in triangles ABC and DEF, angle A = angle D, angle B = angle E, and AB = DE, then triangle ABC is congruent to triangle DEF.

Congruent Triangles

Congruent Triangles
Rules for Triangle Congruency	SSS Rule	SAS Rule
ASA Rule	AAS Rule	Two-column Proofs

Rules for Triangle Congruency

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.

Table of Contents Show

Congruent Triangles
Rules for Triangle Congruency
Side-Side-Side (SSS) Rule
Side-Angle-Side (SAS) Rule
Angle-Side-Angle (ASA) Rule
Angle-Angle-Side (AAS) Rule
Three Ways To Prove Triangles Congruent
Using Two Column Proofs To Prove Triangles Congruent
What are the 4 ways to prove triangles are congruent?
What are 5 ways to prove a triangle is congruent?
How do you know △ ABC ≅ △ DEF explain?

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.

The following diagrams show the Rules for Triangle Congruency: SSS, SAS, ASA, AAS and RHS. Take note that SSA is not sufficient for Triangle Congruency. Scroll down the page for more examples, solutions and proofs.

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.

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 Angle Non-included angle

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

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-side-angle 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.

Three Ways To Prove Triangles Congruent

A video lesson on SAS, ASA and SSS.

SSS Postulate: If there exists a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding sides of the other triangle, the two triangles are congruent.
SAS Postulate: If there exists a correspondence between the vertices of two triangles such that the two sides and the included angle of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.
ASA Postulate: If there exits a correspondence between the vertices of two triangles such that two angles and the included side of one triangle are congruent to the corresponding parts of the other triangle, the two triangles are congruent.

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Using Two Column Proofs To Prove Triangles Congruent

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

Show Video Lesson

Triangle Congruence by SAS
How to Prove Triangles Congruent using the 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.

Show Video Lesson

Prove Triangle Congruence with ASA Postulate
How to Prove Triangles Congruent using the Angle Side Angle Postulate?
If 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.

Show Video Lesson

Prove Triangle Congruence by AAS Postulate
How to Prove Triangles Congruent using the Angle Angle Side Postulate?
If 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.

Show Video Lesson

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What are the 4 ways to prove triangles are congruent?

These four criteria used to test triangle congruence include: Side – Side – Side (SSS), Side – Angle – Side (SAS), Angle – Side – Angle (ASA), and Angle – Angle – Side (AAS). There are more ways to prove the congruency of triangles, but in this lesson, we will restrict ourselves to these postulates only.

What are 5 ways to prove a triangle is congruent?

There are 5 main rules of congruency for triangles:.

SSS Criterion: Side-Side-Side..

SAS Criterion: Side-Angle-Side..

ASA Criterion: Angle-Side- Angle..

AAS Criterion: Angle-Angle-Side..

RHS Criterion: Right angle- Hypotenuse-Side..

How do you know △ ABC ≅ △ DEF explain?

If two angle in one triangle are congruent to two angles of a second triangle, and also if the included sides are congruent, then the triangles are congruent. Using labels: If in triangles ABC and DEF, angle A = angle D, angle B = angle E, and AB = DE, then triangle ABC is congruent to triangle DEF.