Congruence of Triangles


 
 
Concept Explanation
 

Congruence of Triangle

Congruence means that one figure will exactly superimpose the other figure. Two congruent line segments have the same length and conversely two lines segments of equal length are congruent. or we can say that two line segments are congruent if and only if their lengths are equal. Similarly two angles are congruent if and only if their measures are equal, or we can say that two angles BAC and EDF are congruent if m large angle BAC=mangle EDF

Two triangles are congruent if and only if one of them can be made to superpose on the other, so as to cover it exactly. Let large Delta ABC;and;Delta DEF be two congruent triangles. Then, we can superpose large Delta ABC on large Delta DEF, so as to cover it exactly. In such a superposition the vertices of large Delta ABC will fall on the vertices of large Delta DEF, in some order. Let us assume that the vertex A falls on vertex D, vertex B on vertex E and vertex C on vertex F.

Then, side AB falls on DE, BC on EF and CA on FD. Also large angle A superposes on the corresponding angle large angle D, angle B;on;angle E and  large angle C;on;angle F. Thus, the order in which the vertices match, automatically determines a correspondence between the sides and angles of the two triangles. And, if the superposition is exact i.e. the triangles are congruent, the corresponding sides and angles are congruent. Consequently, we get six equalities three of the corresponding sides and three of the corresponding angles. If large Delta ABC superposes on large Delta DEF exactly such that the vertices of large Delta ABC fall on the vertices of large Delta DEF in the following order

   large Aleftrightarrow D,Bleftrightarrow E,Cleftrightarrow F

Then, we have the following six qualities

    large AB=DE,BC=EF,CA=FD    (i.e., corresponding sides are congruent)

 large angle A=angle D,angle B=angle E,angle C=angle F       (i.e., corresponding angles are congruent)

Congruence Relation

From the definition of congruence of two triangles, we obtain the following results:

(i) Every triangle is congruent to itself i.e., large Delta ABCcong Delta ABC

(ii) If large Delta ABCcong Delta DEF, then large Delta DEFcong Delta ABC

(iii) If large Delta DEFcong Delta ABC, and large Delta DEFcong Delta PQR, then large Delta ABCcong Delta PQR

Note: It is important to write the correspondence  of vertices correctly for writing of congruence of triangles in symbolic form because large Delta ABCcong Delta DEFbut large Delta ABCncong Delta EFD and the symbolic form is used to write corresponding parts of congruent triangles {C.P.C.T.}

Sample Questions
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Question : 1

If two triangles  Delta ABC: : and : : Delta PQR  are congruent to  Delta XYZ,  then which of the following is true ?

Right Option : D
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Explanation
Question : 2

If  Delta ABC cong Delta PQR : : and : : Delta ABC is not congruent to  Delta RPQ, then which of the following is not true ?

Right Option : A
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Explanation
Question : 3

In the figure shown above,if bigtriangleup ABD;and;bigtriangleup PQM and Side AB exactly falls on Side PQ, AD exactly falls on Side PM and BD must fall on side _____________ so that the triangles are congruent.

Right Option : A
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Explanation
 
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