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Congruence of Triangles

Triangles I - Concepts

Calculating Perimeter of Triangle

Congruence of Triangles

Congruence of Triangles By SAS Criteria

Congruence of Triangles By ASA Criteria

Congruence of Triangles By AAS Criteria

Class - 9th IMO Subjects

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 $large Delta ABCcong Delta DEF$ but $large Delta ABCncong Delta EFD$ and the symbolic form is used to write corresponding parts of congruent triangles {C.P.C.T.}

Sample Questions

(More Questions for each concept available in Login)

Question : 1

D,E,F are the mid-point of the sides BC, CA and AB respectively of $Delta ABC. : : Then : : Delta DEF$ is congruent to triangle
A. ABC	B. AEF	C. $BFD,CDF$	D. $AFE,BFD,CDE$
Right Option : D
View Explanation

Question : 2

If two triangles $Delta ABC: : and : : Delta PQR$ are congruent to $Delta XYZ,$ then which of the following is true ?
A. $Delta ABC cong Delta PQR$	B. $angle A=angle P$	C. $AB=PQ$	D. All of these
Right Option : D
View Explanation