Congruence Criterion: Side, Side, Side

🏆Practice 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.

Test yourself on side, side, side!

Look at the parallelogram ABCD below.

What can be said about triangles ACD and ABD?

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:

Example 1

Given the triangles $Δ ABC$ and $Δ DEF$ such that

$AB = DE$ (edge)

$BC = EF$ (edge)

$AC = DF$ (edge)

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

We will write it as follows:

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

From this we can also deduce that:

$∠A = ∠D$
$∠B = ∠E$
$∠C = ∠F$

since these are corresponding angles and are equal in congruent triangles

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Example 2: Triangle Congruence Exercise

Given the two triangles $Δ ABC$ and $Δ ACD$ such that $AC$ is the common side.

We are also informed that:

$AB=DA$

$DC=CB$

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

Proof:

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

Let's see

$AC=DC$ (side)

$AB=DB$ (side)

We realize that $AC$ (side) is common to both triangles

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

Consequently, we can deduce that

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

$QED$

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:

$EC=EB$

$AC=AB$

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

Solution

Since $EC=EB$

Since $AC=AB$

Common side $AE=AE$

The triangles are congruent by $SSS$

Congruent by $SSS$

Do you know what the answer is?

Exercise 2

Assignment

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

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

Solution

$AB=AC$

Since triangle $ABC$ is isosceles

$BK=KC$

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

$AK=AK$

Common side

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

Overlap according to $S.S.S$

Exercise 3

Assignment

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

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

Solution

$BE$ and $AC$

Intersect at point $D$

$AD=DC$

$D$ intersects $BE$

$\angle ADB=\angle EDC$

Vertically Opposite Angles

Congruent triangles by $S.A.S$

Congruent by $S.A.S$

Exercise 4

Assignment

The triangles $ΔABC≅ΔEFG$

In triangle $ΔABC$ we draw the median $AD$

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

We demonstrate: $ΔADB≅ΔEHF$

Solution

$AB=EF$

Given that triangles $ΔABC$ and $ΔEFG$ are congruent

$AD=EH$

In congruent triangles, the medians are necessarily equal

(coming from the same vertex to the same base)

$BD=FH$

The median bisects the base it reaches.

Congruent triangles by $S.S.S$

Congruent according to $S.S.S$

Exercise 5

Assignment

Given the isosceles trapezoid $ABCD$.

Inside it contains the square $ABFE$.

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

Solution

$ABCD$ is an isosceles trapezoid (given)

$AD=BC$

Isosceles trapezoid

Since $ABFE$ is a square

$AE=BF$

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

$\sphericalangle D=\sphericalangle C$

The base angles in an isosceles trapezoid are equal

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

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

$\sphericalangle DAE=\sphericalangle FBC$

if two angles are equal then the third is also equal

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

$S.A.S$

Do you think you will be able to solve it?

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 lengths in their sides and likewise the same degrees 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.

examples with solutions for side, side, side

Exercise #1

Look at the parallelogram ABCD below.

What can be said about triangles ACD and ABD?

Step-by-Step Solution

According to the side-angle-side theorem, the triangles are similar and coincide with each other:

AC = BD (Any pair of opposite sides of a parallelogram are equal)

Angle C is equal to angle B.

AB = CD (Any pair of opposite sides of the parallelogram are equal)

Therefore, all of the answers are correct.