Class 09 To verify experimentally the different criteria for congruency of triangles using triangle cut-outs.

 Activity 14

 

 

 

OBJECTIVE                                                                  

 

To verify experimentally the different criteria for congruency of triangles using triangle cut-outs.

   MATERIAL REQUIRED

 

Cardboard, scissors, cutter, white paper, geometry box, pencil/sketch pens, coloured glazed papers.

 

METHOD OF CONSTRUCTION

 

1.   Take a cardboard of a convenient size and paste a white paper on it.

 

2.   Make a pair of triangles ABC and DEF in which AB = DE, BC = EF, AC = DF on a glazed paper and cut them out [see Fig. 1].

 

Make a pair of triangles GHI, JKL in which GH = JK, GI = JL, ∠G = ∠J on a glazed paper and cut them out [see Fig. 2].

4.   Make a pair of triangles PQR, STU in which QR = TU, ∠Q = ∠T, ∠R = ∠U on a glazed paper and cut them out [see Fig. 3].

 

5.    Make two right triangles XYZ, LMN in which hypotenuse YZ = hypotenuse MN and XZ = LN on a glazed paper and cut them out [see Fig. 4].

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 3

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

 

Fig. 4

 

DEMONSTRATION

 

1.   Superpose DABC on DDEF and see whether one triangle covers the other triangle or not by suitable arrangement. See that ΔABC covers ΔDEF completely only under the correspondence A↔D, B↔E, C→F. So, ΔABC ≅ ΔDEF, if AB = DE, BC = EF and AC = DF.

 

This is SSS criterion for congruency.

2.   Similarly, establish ΔGHI ≅ ΔJKL if GH = JK. ∠G = ∠J and GI = JL. This is SAS criterion for congruency.

 

3.   Establish ΔPQR ≅ ΔSTU, if QR = TU, ∠Q = ∠T and ∠R = ∠U. This is ASA criterion for congruency.

 

4.   In the same way, ΔSTU ≅ ΔLMN, if hypotenuse YZ = hypotenuse MN and XZ = LN.

 

This is RHS criterion for right triangles.

OBSERVATION
On actual measurement :
In ΔABC and ΔDEF,
AB = DE = ...................	,	BC = EF = ...................	,
AC = DF = ...................	,	∠A = ...................	,
∠D = ...................	,	∠B = ...................	,
∠C = ...................	,	∠F = ....................
Therefore, ΔABC ≅ ΔDEF.
2. In ΔGHI and ΔJKL,
GH = JK = ...................	,	GI = JL = ....................	,
KL= ...................	,	∠G = ...................	,
∠H = ...................	,	∠K = ...................	,
∠L = ....................
Therefore, ΔGHI ≅ ΔJKL.
3. In ΔPQR and ΔSTU,
QR = TU = ...................	,	PQ = ...................	,
PR = ...................	,	SU = ....................
∠Q = ∠T = ...................	,	∠R = ∠U = ...................	,

∠E = ...................,

 

 

 

 

 

 

HI =	...................,
∠J =	...................,
∠I =	...................,

 

 

 

 

 

 

ST = ...................,

 

∠S = ...................,

 

∠P = ....................

4. In ΔXYZ and ΔLMN, hypotenuse YZ = hypotenuse MN = .............
XZ=LN=	...................,	XY = ...................	,
LM = ...................	,	∠X=∠L=90°
∠Y = ...................	,	∠M = ...................	,	∠Z = ...................	,
∠N = ...................	,

Therefore, ΔXYZ ≅ ΔLMN.

 

APPLICATION

 

These criteria are useful in solving a number of problems in geometry.

 

These criteria are also useful in solving some practical problems such as finding width of a river without crossing it.