What is an example of the SSS postulate/theorem?

Two triangles are said to be congruent if they have same shape and same size. When triangles are congruent corresponding sides (sides in same position) and corresponding angles (angles in same position) are congruent (equal).

There are two theorems and three postulates that are used to identify congruent triangles.

Angle-Angle-Side Theorem (AAS theorem)

As per this theorem the two triangles are congruent if two angles and a side not between these two angles of one triangle are congruent to two corresponding angles and the corresponding side not between the angles of the other triangle.

Example:

Hypotenuse-Leg Theorem (HL theorem)

If the hypotenuse and one of the legs (sides) of a right triangle are congruent to hypotenuse and corresponding leg of the other right triangle, the two triangles are said to be congruent.

Side-Side-Side Postulate (SSS postulate)

If all three sides of a triangle are congruent to corresponding three sides of other triangle then the two triangles are congruent.

Angle-Side-Angle Postulate (ASA postulate)

According to this postulate the two triangles are said to be congruent if two angles and the side between these two angles of one triangle are congruent to corresponding angles and the included side (side between two angles) of the other triangle.

Side-Angle-Side Postulate (SAS postulate)

If two sides and the included angle (angle between these two sides) of one triangle are congruent to the corresponding two sides and the included angle of a second triangle, then the two triangles are congruent.

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Side-Side-Side or SSS is a kind of triangle congruence rule where it states that if all three sides of one triangle are equal to all three corresponding sides of another triangle, the two triangles are considered to be congruent. Two or more triangles are said to be congruent when the measurements of the corresponding sides and the corresponding angles are equal. Let us learn more about the SSS congruence rule, SSS theorem, the formula, and solve a few examples.

Definition of SSS

SSS Criterion stands for Side-Side-Side congruence postulate. Under this criterion, if all the three sides of one triangle are equal to the three corresponding sides of another triangle, the two triangles are said to be congruent. This essentially means that any such pair of triangles will be equiangular i.e. all corresponding angle pairs are equal. Look at the triangles below, the triangles are said to be congruent because AC = ZY, CB = ZX, and AB = XY. Hence, ∆ABC ≅ ∆XYZ.

Congruence of Triangle

The word congruent means equal in every aspect or figure in terms of shape and size. Congruence is the term used to describe the relation of two figures that are congruent. For example, the two triangles above are said to be congruent according to the SSS congruence rule. This means that AB falls on XY, BC falls on XZ, and AC falls on ZY. Hence, it indicates that the corresponding parts of congruent triangles are equal. Since these two triangles are of the same size and shape, we can say that they are congruent. We can represent this in a mathematical form using the congruent triangles symbol (≅).

Proof of SSS Congruence Rule

SSS or Side-Side-Side congruence rule states that if three sides of one triangle are equal to three corresponding sides of another triangle, then the triangles are congruent. Let's perform an activity to show SSS proof. Draw two right-angled triangles with the hypotenuse of 6 inches and one side of 4 inches each. Cut these triangles and try to place one triangle over the other such that equal sides are placed over one another. Do you observe that these two triangles superimpose on each other completely? This means these two triangles are congruent. This completes the SSS proof.

SSS or Side-Side-Side Similarity Criterion

According to the SSS similarity theorem, two triangles will the similar to each other if the corresponding ratio of all the sides of the two triangles are equal. This criterion or rule is commonly used when we only have the measure of the sides of the triangle and have less information about the angles of the triangle.

In the image given below, if it is known that PQ/ED = PR/EF = QR/DF

And we can say that by the SSS similarity criterion, △PQR and △EDF are similar or △PQR ≅ △EDF.

Side Side Side Congruence Theorem

Side-Side-Side (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.

SSS Formula

Using the SSS Formula, the congruency or similarity of any two triangles can be checked when two sides and the angle between these sides for both the triangles follow the required criterion. There are different SSS triangle formulas used to prove the congruence or similarity between two triangles.

SSS Congruence Rule

The Side-Side-Side congruence rule states that, if all the three sides of a triangle are equal to the three sides of another triangle then the triangles are congruent.

SSS Similarity Rule

The SSS similarity criterion states that if the three sides of one triangle are respectively proportional to the three sides of another, then the two triangles are similar.

1. Example 1: The two points P and Q are on the opposite sides of the line segment AB. The points P and Q are equidistant from points A and B. Can you prove that ∆PAQ is congruent to the ∆PBQ?

   

   Solution:

   As the two points, P and Q are equidistant from the endpoints of the line segment AB. Therefore,

   AP =BP and AQ = BQ

   Now the side PQ is common in both the triangles ∆PAQ and ∆PBQ. Therefore according to the SSS postulate, the two triangles are congruent.

   Hence, ∆PAQ ≅ ∆PBQ

2. Example 2: Triangle ABC is an isosceles triangle and the line segment AD is the angle bisector of the angle A. Can you prove that ∆ADB is congruent to the ∆ADC?

   

   Solution:

   The triangle ABC is an isosceles triangle, therefore, AB=AC. Now the side AD is common in both the triangles ∆ADB and ∆ADC. As the line segment AD is the angle bisector of the angle A then it divides the line segment BC into two equal parts BD and CD. Therefore,

   BD =CD and AB = AC

   Now according to the SSS postulate, the two triangles are congruent.

   Hence, ∆ADB ≅ ∆ADC

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FAQs on SSS

SSS Criterion stands for side side side congruence postulate. Under the SSS theorem, if all the three sides of one triangle are equal to the three corresponding sides of another triangle, the two triangles are congruent.

How Do You Prove SSS Similarity?

SSS similarly can be proved by showing that the side lengths of one triangle are proportional to the side length of the other triangle.

What is the Difference Between SAS and SSS?

Both SAS and SSS rules are the triangle congruence rules. SAS full form is "side-angle-side" and SSS full form is "side-side-side."

• In the SAS postulate, two sides and the angle between them in a triangle are equal to the corresponding two sides and the angle between them in another triangle.
• In the SSS postulate, all three sides of one triangle are equal to the three corresponding sides of another triangle.

What are the Three Triangle Similarity Theorems?

The three triangle similarity theorems are:

1. Angle-Angle (AA)
2. Side-Angle-Side (SAS)
3. Side-Side-Side (SSS)

What is the Side Side Side Rule?

The SSS rule states that, if three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent. The rule helps in proving if the triangles are congruent or not.

What is SSS, SAS, ASA, and AAS?

The 4 different triangle congruence theorems are:

• SSS(Side-Side-Side): Where three sides of two triangles are equal to each other.
• SAS(Side-Angle-Side): Where two sides and an angle included in between the sides of two triangles are equal to each other.
• ASA(Angle-Side-Angle): Where two angles along with a side included in between the angles of any two triangles are equal to each other.
• AAS(Angle-Angle-Side): Where two angles of any two triangles along with a side that is not included in between the angles, are equal to each other.