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.

Clas 09 the vertically opposite angles are equal (ii) the sum of two adjacent angles is 180º (iii) the sum of all the four angles is 360º.

 

Activity 13

OBJECTIVE                                                                   

To verify experimentally that if two lines intersect, then

 (i)       the vertically opposite angles are equal

 (ii)        the sum of two adjacent angles is 180º

 (iii)      the sum of all the four angles is 360º.

  MATERIAL REQUIRED

Two transparent strips marked as AB and CD, a full protractor, a nail, cardboard, white paper, etc.

METHOD OF CONSTRUCTION

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

2.   Paste a full protractor (0° to 360º) on the cardboard, as shown in Fig. 1.

3.   Mark the centre of the protractor as O.

4.   Make a hole in the middle of each transparent strip containing two intersecting lines.

5.   Now fix both the strips at O by putting a nail as shown in Fig. 1.

DEMONSTRATION

1.   Observe the adjacent angles and the vertically opposite angles formed in different positions of the strips.

 2.   Compare vertically opposite angles formed by the two lines in the strips in different positions.

 3.   Check the relationship between the vertically opposite angles.

 4.   Check that the vertically opposite angles ∠AOD, ∠COB, ∠COA and ∠BOD are equal.

 5.   Compare the pairs of adjacent angles and check that ∠COA + ∠DOA= 180º, etc.

 6.   Find the sum of all the four angles formed at the point O and see that the sum is equal to 360º.

 OBSERVATION

 On actual measurement of angles in one position of the strips : 

1.	∠AOD = .................	, ∠AOC = ...................
	∠COB = .................	, ∠BOD = .................
Therefore, ∠AOD = ∠COB and ∠AOC = ............			(vertically opposite angles).
2.	∠AOC + ∠AOD = .............	, ∠AOC + ∠BOC =		...................,
	∠COB + ∠BOD = ...................
	∠AOD + ∠BOD = ...................	(Linear pairs).
3.	∠AOD + ∠AOC + ∠COB + ∠BOD = ....................			(angles formed at a point).

 APPLICATION

 These properties are used in solving many geometrical problems.

Class 09 To find a hidden picture by plotting and joining the various points with given coordinates in a plane.

 Activity 12

OBJECTIVE                                                                    

To find a hidden picture by plotting and joining the various points with given coordinates in a plane.

 MATERIAL REQUIRED

Cardboard, white paper, cutter, adhesive, graph paper/squared paper, geometry box, pencil.

METHOD OF CONSTRUCTION

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

2.   Take a graph paper and paste it on the white paper.

3.   Draw two rectangular axes X′OX and Y′OY as shown in Fig. 1.

4.   Plot the points A, B, C, ... with given coordinates (a, b), (c, d), (e, f), ..., respectively as shown in Fig. 2.

Join the points in a given order say A→B→C→D→.....→A [see Fig. 3]

DEMONSTRATION

By joining the points as per given instructions, a ‘hidden’ picture of an ‘aeroplane’ is formed.

OBSERVATION

In Fig. 3:

Coordinates of points A, B, C, D, .......................

are ........, ........, ........, ........, ........, ........, ........

Hidden picture is of ______________.

APPLICATION

This activity is useful in understanding the plotting of points in a cartesian plane which in turn may be useful in preparing the road maps, seating plan in the classroom, etc.

Class 09 To find the values of abscissca and ordinates of various points given in a cartesian plane.

 Activity 11

  OBJECTIVE                                                                   

To find the values of abscissca and ordinates of various points given in a cartesian plane.

  MATERIAL REQUIRED

 

Cardboard, white paper, graph paper with various given points, geometry box, pen/pencil.

 METHOD OF CONSTRUCTION

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

 2.    Paste the given graph paper alongwith various points drawn on it [see Fig. 1].

 3.   Look at the graph paper and the points whose abcissae and ordinates are to be found.

 DEMONSTRATION

 To find abscissa and ordinate of a point, say A, draw perpendiculars AM and AN from A to x-axis and y-axis, respectively. Then abscissa of A is OM and ordinate of A is ON. Here, OM = 2 and AM = ON = 9. The point A is in first quadrant. Coordinates of A are (2, 9)

OBSERVATION

 

Point

Abscissa

Ordinate

Quadrant

Coordinates

 

B

 

C

 

...

 

...

 

...

 

...

		PRECAUTION
	Fig. 1	The students should be careful
APPLICATION		while reading the coordinates,

 

This activity is helpful in locating the position	otherwise the location of the
	object will differ.

of a particular city/place or country on map