Sss Congruence Theorem

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If there is a correspondence between the vertices of two triangles such that three sides of one triangle are congruent to the corresponding three sides of the other triangle, then that correspondence is a congruence.

In the above diagram, if three sides of one triangle are congruent to the three sides of the second triangle, then the triangles are to be congruent, and the type of the congruence in the above diagram is known as SSS or Side-Side-Side.

Properties of sss triangle congruence

SSS is generally the form of Side-Side-Side Triangle Congruence Theorem (or may be called off Edge-Edge-Edge Triangle Congruence Theorem).

Proof for the sss triangle congruence:

An one approach of proving two angles are same by sss is that,

start flipping of the triangle DEF vertically, keeping the vertex D at vertex A.

considering the measure of the lines AC and DF are equal, the vertex F also should consconce on vertex C.

The transformed shape is shown in the following diagram.

After superimposing DF on AC,Join the line BE which intersects AC (or DF) at G.

Hence, ...
... AC or DF is perpendicular and bisects BE.

This means BG = EG

From that

Example :

In the following figure, it is given that UV = WX and UX = VW. Prove that Δ UXW is congruent to Δ WVU.

Solution: In Δ UXW and Δ WVU, we have,

UV = WX ( given )

UX = VW ( given)

UW = UW ( common side)

So, by SSS congruence we get,

Δ UXW is congruent to Δ WVU

Examples for sss triangle congruence

Example:

Given Δ UVW with vertices U(0, 3), V(2, 0), and W(0, 0) and Δ XYZ with vertices X(5, 6), Y(5, 3), Z(3, 3). Show that ΔUWV ≅ Δ XYZ.

Solution:

By using Side-Side-Side property:

If the sides of first triangle given are congruent to the sides of a second triangle given, then the triangles are congruent.

The Side-Side-Side postulate can be used to prove triangles congruent.

Using the distance formula to show that the corresponding sides are congruent.

UW = √ (0-0)2 + (3-0)2

UW =√9 = 3

XY = √(5 − 5)2 + (6− 3)2

XY = √9 = 3

UV = √(0 − 2)2+ (5 − 0)2

UV =√29

XZ = √(5 − 3)2 + (8 − 3)2

XZ = √29

WV = √(0 − 2)2 + (0 − 0)2

WV =√4 = 2

YZ = √(5 − 3)2 + (3 − 3)2

YZ =√4 = 2

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In the given two triangle all the pairs of corresponding sides are congruent, so ΔUWV ≅ Δ XYZ by Side-Side-Side

“Two triangles are congruent if three sides of one triangle are equal to the corresponding three sides of the other triangles in SSS congruence theorem proof”

Given:

In Deltas ABC and DEF,

AB = DE, BC = EF and AC = DF.

Result.

DeltaABC ~= angleDEF

angleA = angleD, angleB = angleE and angleC = angleF.

SSS CONGRUENCE THEOREM

Characteristic property of SSS Congruence theorem:

Statement:

In SSS congruence theorem, if the corresponding sides of two triangles are proportional, then they are similar.

The theorem of the above property is discussed in the form of the following solution.

Solution:

Given: In DeltaABC and DeltaDEF, we are given that

(AB)/(DE) = (BC)/(EF) = (AC)/(DF)

To prove:

DeltaABC ~ DeltaDEF.

Construction: mark point P on DE and the point Q on DF such that DP = AB, DQ = AC. Join P and Q.

Characteristic of SSS Congruence theorem

Theorem:

Since (AB)/(DE) = (AC)/(DF)

(given)

(DP)/(DE) = (DQ)/(EF)

(construction)

By the converse of Thales theorem, PQ || EF.

anlgeDPQ = angleE

(corresponding angles)

angleDQP = angleF

(corresponding angles)

By AAA similarity ( or by AA similarity)

We get DeltaDPQ ~ DeltaDEF

(DP)/(DE) = (PQ)/(EF) rArr (AB)/(DE) = (PQ)/(EF)
(given)

But (AB)/(DE) = (BC)/(EF)
(given)

BC = PQ and AB = DP, AC = DQ
(construction)

DeltaABC ~= DeltaDPQ
(SSS)

Since

DeltaDPQ ~ DeltaDEF

DeltaABC ~ DeltaDEF.

Hence, the required result is proved in SSS congruence theorem.

Proportional of SSS Congruence Theorem:

As we have said, proportional and similar denote exactly the identical thing. However, because the word proportional has the word proportion (two fractions located equal to each other), the term proportional may assist the reader start on to set up proportions between similar triangles in SSS congruence theorem proof.

Take the following two triangles, for example:

Proportional of SSS Congruence Theorem

Because the triangles are similar or proportional, we can set up a proportion to find x, see if you can follow the pattern:

5/10 = 6/x

( the first step in solving a proportion is to cross multiply)

5x=6xx10

5x=60

x=60/5

x=12

This answer totally makes sense, as it can be see that each side doubled from the smaller triangle to the larger triangle.

Learn more on about Define Scalene Triangle and its Examples. Between, if you have problem on these topics online algebra 2 help, Please share your comments.