Congruence of Triangles and Geometric Figures

Two triangles are said to be congruent when all three corresponding sides are equal and all three corresponding angles are identical in measure. These triangles can be moved around, rotated, flipped, and turned to make them seem the same. They will line up if they are moved. The symbol of congruence is’ ≅’. Congruent triangles have the same sides and angles as each other. There are basically four congruence rules that are rhs sss sas asa used to  prove if two triangles are congruent.However, all six dimensions must be discovered. As a result, the congruence of triangles may be determined using only three of the six variables.

In this article we will learn  rhs congruent triangles and other congruent triangles sss sas asa rhs along with solved examples.

Conditions of Congruence in Triangles

Two triangles are said to be congruent if both the triangles have similar size and shape. It is not necessary to find all six corresponding elements of both triangles to be equal in order to determine that they are congruent. There are five prerequisites for two triangles to be congruent, according to studies and experiments. They are sss sas asa rhs and aas congruence properties.

Congruent Triangles

Both triangles are said to be congruent if their three angles and three sides are equivalent to the equivalent angles and sides of another triangle. In the Δ PQR and ΔXYZ,We can see that PQ = XY, PR = XZ, and QR = YZ in the PQR and XYZ, and that  ∠P = ∠ X,  ∠Q = ∠ Y, and also ∠R =  ∠Z. Hence we can say that Δ PQR ≅ ΔXYZ. 

[Image will be Uploaded Soon]

To be congruent, the two triangles must be the same size and shape. Both of the given triangles should be superimposed on one another. A triangle's location or appearance appears to change when we rotate, reflect, and/or translate it. In that instance, we must determine the six parts of a triangle as well as the parts of the other triangle that correspond to them. Consider Δ ABC and ΔPQR as shown below.

Identifying the Corresponding Parts

[Image will be Uploaded Soon]

Thus on identifying the corresponding parts of the given triangles, we say that the Δ ABC ≅ ΔPQR.

CPCT

We come across the word CPCT, when we study about the congruent triangle. CPCT means “Corresponding Parts of Congruent Triangles”. The matching parts of congruent triangles are equal, as we know. We typically utilise the abbreviation cpct in brief phrases instead of the complete version while dealing with triangle principles and solving problems.

We can predict the congruence without actually measuring the sides and angles of a triangle. Different rules of congruency are as follows.

SSS (Side-Side-Side)

SAS (Side-Angle-Side)

ASA (Angle-Side-Angle)

AAS (Angle-Angle-Side)

RHS (Right angle-Hypotenuse-Side)

RHS Congruent Triangles

Here we will discuss congruence rhs sss and other congruence methods of triangle.

According to congruent triangles rhs rule : In two right-angled triangles, when the length of the hypotenuse and corresponding side of one triangle equals the length of the hypotenuse and corresponding side of the second triangle, the two triangles are congruent. It is also known as rhs triangle congruence.

[Image will be Uploaded Soon]

We can see in above figure, hypotenuse XZ = RT and side YZ=ST

Hence triangle XYZ ≅ triangle RST.

SSS Congruence Rule

According to the theorem: The two triangles are congruent if the three sides of one triangle are equal to the corresponding three sides (SSS) of the other triangle

[Image will be Uploaded Soon]

We can see in the above-given figure, AB= PQ, QR= BC and AC=PR

Hence Δ ABC ≅ Δ PQR.

ASA ( Angle - Side - Angle ) Congruence Rule

The two triangles are said to be congruent if any two angles and the side included between the angles of one triangle are comparable to the corresponding two angles and side included between the angles of the second triangle. Hence the two triangles are said to be congruent by ASA rule.

[Image will be Uploaded Soon]

We can see in above figure, ∠ B = ∠ Q, ∠ C = ∠ R and the sides between ∠B =∠C , ∠Q =∠ R  i.e. BC= QR. Hence, Δ ABC ≅ Δ PQR.

SAS (Side-Angle-Side)

According to the SAS theorem - The two triangles are said to be congruent if any two sides and the angle included between the sides of one triangle are equivalent to the corresponding two sides and the angle between the sides of the another triangle.

[Image will be Uploaded Soon]

We can see in above given figure, length of sides AB= PQ, AC=PR and angle between AC and AB equal to angle between PR and PQ i.e. ∠A = ∠P. Hence, Δ ABC ≅ Δ PQR.

Angle - Angle - Side( AAS)

According to the Angle - Angle - Side rule :  Two triangles are congruent if their corresponding two angles and one non-included side are equal.

[Image will be Uploaded Soon]

Given that:

∠ BAC = ∠ QPR, ∠ ACB = ∠ RQP and AB=QR,

So the triangle ABC and PQR are congruent to each other (△ABC ≅△ PQR).

Solved Examples:

1. In the Figure Given Below, AB = BC and AD = CD. Show that BD Bisects AC at Right Angles.

[Image will be Uploaded Soon]

Sol: Consider ∆ABD and ∆CBD,

Given AB = BC                                                 

AD = CD                                                

Common Side BD = BD                                               

By SSS congruency Therefore, ∆ABD ≅ ∆CBD        

(By CPCT)

∠ABD = ∠CBD                                  

Now, consider ∆ABE and ∆CBE,

(Given) AB = BC                                               

∠ABD = ∠CBD                                     (Proved above)

Common Side BE = BE                                            

Therefore, ∆ABE≅ ∆CBE                  (By SAS congruency)

∠BEA = ∠BEC                                     (CPCTC)

According to Linear pair 

∠BEA +∠BEC = 180°                

2∠BEA = 180°                                    (∠BEA = ∠BEC)

∠BEA = 180°/2 = 90° = ∠BEC

By CPCT AE = EC       

Hence, BD is a perpendicular bisector of AC.

2. In the Given Triangle Figures, Prove that Two Triangles are Congruent.

[Image will be Uploaded Soon]

Sol: To prove two triangles are congruent we need to find value of ∠ABC

Given ∠BAC = 65° and ∠BCA = 55°

In ∆ABC, ∠BAC + ∠ABC + ∠BCA = 180°

⟹ 65° + ∠ABC +55° = 180°

⟹ ∠ABC = 60°.

We can see in both  ∆ABC and ∆XYZ,

Gien AB = XZ = 4 cm, BC = YZ = 5 cm 

Also we found value of ∠ABC

So, ∠ABC = ∠XZY = 60°.

Therefore, both the triangles are congruent by SAS (Side-Angle-Side) criterion of congruence.

Conclusion:

We have discussed different types of congruence methods of triangles. We have learnt congruence rhs sss and rhs triangle congruence. From above we can conclude that following are the important points that we need to remember about congruence of triangle:

• If the six parts of one triangle are equal to the corresponding six parts of the other triangle, the two triangles are congruent..

• There are five criteria used to determine triangle congruence. Five conditions are SSS, SAS, ASA, AAS, and RHS criteria.

• Two triangles that have equal corresponding angles may not be congruent to each other because one triangle may be an enlarged copy of the another triangle. Hence, there is no AAA Criterion for Congruence.

• Triangles with corresponding sides and angles are said to be congruent. Congruence is denoted by the symbol “≅”.  Both the triangles have the same area as well as the same perimeter