Class 09 To verify that the sum of the angles of a triangle is 180º.

 Activity 15

  OBJECTIVE                                                                    

To verify that the sum of the angles of a triangle is 180º.

 METHOD OF CONSTRUCTION

 MATERIAL REQUIRED

Hardboard sheet, glazed papers, sketch pens/pencils, adhesive, cutter, tracing paper, drawing sheet, geometry box.

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

 2.   Cut out a triangle from a drawing sheet, and paste it on the hardboard and name it as ΔABC.

 3.   Mark its three angles as shown in Fig. 1

 Cut out the angles respectively equal to ∠A, ∠B and ∠C from a drawing sheet using tracing paper [see Fig. 2].

5.   Draw a line on the hardboard and arrange the cut-outs of three angles at a point O as shown in Fig. 3.

 Fig. 3

 DEMONSTRATION

 The three cut-outs of the three angles A, B and C placed adjacent to each other at a point form a line forming a straight angle = 180°. It shows that sum of the three angles of a triangle is 180º. Therefore, ∠A + ∠B + ∠C = 180°.

 OBSERVATION

 Measure of ∠A = -------------------.

 Measure of ∠B = -------------------.

 Measure of ∠C = -------------------.

 Sum (∠A + ∠B + ∠C) = -------------------.

 APPLICATION

 This result may be used in a number of geometrical problems such as to find the sum of the angles of a quadrilateral, pentagon, etc

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.