Congruence Criterion: Side, Side, Side

There are 4 criteria to determine that two triangles are congruent. In this article, we will learn to use the third criterion of congruence:

Side, Side, Side

Definition: 2 triangles in which their three sides are of the same length are congruent triangles.

In this article, we will study this criterion and see examples of how to apply it.

image Side_Side_Side

Definition:

Two triangles in which all three sides are of the same length are congruent triangles.

To prove that two triangles are congruent we can use one of the following postulates:

  • SAS - side, angle, side
  • ASA - angle, side, angle
  • SSS - side, side, side
  • SSA - side, side, angle

Example 1

Given the triangles ΔABCΔ ABC and ΔDEFΔ DEF such that

AB=DEAB = DE (edge)

BC=EFBC = EF (edge)

AC=DFAC = DF (edge)

Side, Side, Side

Therefore, we can deduce that: ΔABCΔ ABC and ΔDEFΔ DEF are congruent triangles according to the Side, Side, Side congruence criterion.

We will write it as follows:

ΔDEFΔABCΔ DEF ≅ Δ ABC according to the congruence criterion: Side, Side, Side (SSS)

From this we can also deduce that:

A=D∠A = ∠D
B=E∠B = ∠E
C=F∠C = ∠F

since these are corresponding angles and are equal in congruent triangles


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Example 2: Triangle Congruence Exercise

Given the two triangles ΔABCΔ ABC and ΔACDΔ ACD such that ACAC is the common side.

Given the two triangles 2

We are also informed that:

AB=DAAB=DA

DC=CBDC=CB

Prove that the triangles ΔABC ΔABC and ΔACD ΔACD are congruent triangles.

Proof:

We will base our proof on the criterion we just learned.

Let's see

AC=DC AC=DC (side)

AB=DB AB=DB (side)

We realize that ACAC (side) is common to both triangles

From this, it follows that in both triangles ΔABCΔ ABC and ΔADCΔ ADC there are three pairs of equal sides.

Consequently, we can deduce that

ΔADCΔABCΔ ADC ≅ Δ ABC according to the Side, Side, Side congruence criterion.

QED QED


If you're interested in this article, you might also be interested in the following articles:

Congruence Criterion: Side, Angle, Side

Congruence Criterion: Angle, Side, Angle

Side, Side, and the Angle Opposite the Larger of the Two Sides

The Method of Writing a Formal Proof in Geometry

On the Tutorela blog, you'll find a variety of articles on mathematics.


Congruence Criteria Exercises: Side, Side, Side

Exercise 1

Assignment

In the given figure:

EC=EB EC=EB

AC=AB AC=AB

By what theorem are the triangles ΔABEΔACE ΔABE≅ΔACE congruent?

In the given figure EC=EB AC=AB

Solution

Since EC=EB EC=EB

Since AC=AB AC=AB

Common side AE=AE AE=AE

The triangles are congruent by SSS SSS

Answer

Congruent by SSS SSS


Exercise 2

Assignment

In an isosceles triangle ABC \triangle ABC we draw the height AK AK .

According to which theorem of congruence do the triangles ΔABKΔACK ΔABK≅ΔACK overlap?

In an isosceles triangle ABC we draw the height AK

Solution

AB=AC AB=AC

Since triangle ABC ABC is isosceles

BK=KC BK=KC

In an isosceles triangle, the height is also a median, and a median cuts the base into two equal parts.

AK=AK AK=AK

Common side

The triangles overlap according to S.S.S S.S.S

Answer

Overlap according to S.S.S S.S.S


Exercise 3

Assignment

The segments BE BE and AC AC intersect at point D D .

Which congruence theorem explains why the triangles ΔABDΔCED ΔABD≅ΔCED are congruent?

The segments BE and AC intersect at point D

Solution

BE BE and AC AC

Intersect at point D D

AD=DC AD=DC

D D intersects BE BE

ADB=EDC \angle ADB=\angle EDC

Vertically Opposite Angles

Overlapping triangles by S.A.S S.A.S

Answer

Overlapping by S.A.S S.A.S


Exercise 4

Assignment

The triangles ΔABCΔEFG ΔABC≅ΔEFG

In triangle ΔABC ΔABC we draw the median AD AD

and in triangle ΔEFG ΔEFG we draw the median EH EH .

We demonstrate: ΔADBΔEHF ΔADB≅ΔEHF

The triangles ΔABC≅ΔEFG

Solution

AB=EF AB=EF

Given that triangles ΔABC ΔABC and ΔEFG ΔEFG are congruent

AD=EH AD=EH

In congruent triangles, the medians are necessarily equal

(coming from the same vertex to the same base)

BD=FH BD=FH

The median bisects the base it reaches.

Congruent triangles by S.S.S S.S.S

Answer

Overlaid according to S.S.S S.S.S


Exercise 5

Assignment

Given the isosceles trapezoid ABCD ABCD .

Inside it contains the square ABFE ABFE .

According to which theorem are the triangles ΔADEΔBCF ΔADE≅ΔBCF congruent?

Given the isosceles trapezoid ABCD. Inside it contains the square ABFE

Solution

ABCD ABCD is an isosceles trapezoid (given)

AD=BC AD=BC

Isosceles trapezoid

Since ABFE ABFE is a square

AE=BF AE=BF

Since ABFE ABFE is a square and all sides in a square are equal

D=C \sphericalangle D=\sphericalangle C

The base angles in an isosceles trapezoid are equal

AED=BFC=90° \sphericalangle AED=\sphericalangle BFC=90°

In a square, all angles are right angles and measure 90° 90° degrees

DAE=FBC \sphericalangle DAE=\sphericalangle FBC

if two angles are equal then the third is also equal

The triangles are congruent according to S.A.S S.A.S

Answer

S.A.S S.A.S


Review Questions

What is the congruence criterion for two triangles?

There are four triangle congruence criteria, which allow us to determine if two triangles have the same dimensions in their sides and likewise the same length in their corresponding angles. In this way, we can say that the two triangles, even when they are in different positions or orientations, will have the same shape and size.


What is the SSS congruence criterion?

This criterion allows us to deduce if two triangles have the same shape and size. According to this criterion, two triangles are congruent when their three sides are equal.


What is the difference between the SSS congruence criterion and the SSS similarity criterion?

The SSS congruence criterion tells us that if two triangles have their three sides equal (congruent sides), then the two triangles are identical, meaning they have the same measurements in terms of sides and angles. Whereas the SSS similarity criterion tells us that if two triangles are similar, then their three sides are proportional, meaning they do not have the same measurement but they do have some proportion between them and they have the same shape, but with different measurements in terms of their sides.


Which pair of triangles are similar by the SSS criterion?

Two triangles will be similar when they have the same shape, regardless of orientation, that is, their corresponding angles are equal but their corresponding sides do not necessarily have the same length, instead, they must have a proportion between them.


What are the criteria for similarity and congruence of triangles?

Congruence criteria

The four triangle congruence criteria are:

  • SAS - Side, Angle, Side.
  • ASA - Angle, Side, Angle.
  • SSS - Side, Side, Side.
  • SSA - Side, Side, Angle.

Similarity criteria

Unlike the congruence criteria, there are only three triangle similarity criteria:

  • SSS - Side, Side, Side.
  • SAS - Side, Angle, Side.
  • AAA - Angle, Angle, Angle.