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VII
Student’
s Material
Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India
CBSE-i
Student’
s Material
CLASS
VII
UNIT-10
Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India
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Preface
Acknowledgment
1.
Syllabus
1
2.
Study Material
2
3.
Student's Support Material
33
l
SW
34
1: Warm Up Activity (W 1)
Shapes
l
SW
2: Warm Up Activity (W 2)
35
Tessellation
l
SW
3: Pre Content Worksheet (P1)
37
Congruence
l
SW
4: Content Worksheet (CW 1)
40
Congruent Figures
l
SW
5: Content Worksheet (CW 2)
42
Congruence of Triangles (SSS)
l
SW
6: Content Worksheet (CW 3)
45
Construction of Triangle (SSS)
l
SW
7: Content Worksheet (CW 4)
46
Congruence of Triangles (SAS)
l
SW8:
Content Worksheet (CW 5)
48
Construction of Triangle (SAS)
l
SW9:
Content Worksheet (CW 6)
49
Congruence of Triangles (ASA)
l
SW10:
Content Worksheet (CW 7)
52
Construction of Triangles (ASA)
l
SW11:
Content Worksheet (CW 8)
Congruence of Triangles (RHS)
53
l
SW12:
Content Worksheet (CW 9)
55
Construction of Triangle (RHS)
l
SW13:
Content Worksheet (CW 10)
56
Construction of Parallel Lines
l
SW
14: Post Content Worksheet (PCW 1)
58
Skill Drill -1
l
SW
15: Post Content Worksheet (PCW 2)
63
Skill Drill -2
4.
Suggested videos/ links/ PPT's
67
Syllabus
Introduction to the concept
Introducing the concept of congruence through
examples of superimposition. Extend congruence to
simple geometrical shapes e.g. triangles, circles
Properties of congruence
Introduction to the properties of congruence of
triangles viz. SSS, SAS, ASA, RHS. Verification of the
properties.
Constructions
Constructions using scale, protractor and compass:
Construction of a line parallel to a given line and from
a point outside it (Simple proof as remark with the
reasoning of alternate angles)
Constructions of simple triangles.
Please note: constructions to be included simultaneously in chapters where they appear.
1
Study
Material
2
Congruence
Introduction
In Unit 7, you have studied about triangles and some of their properties such as angle
sum property, exterior angle property, triangle inequality property and Pythagoras
theorem. In this unit, we extend this study to congruence of plane figures in general and
congruence of triangles, in particular. Further, we shall learn to construct triangles with
given three parts and use these constructions in arriving at some congruence criteria for
two triangles such as SSS, SAS, ASA and RHS. We shall also learn to solve simple
geometric problems using these congruence criteria of triangles.
1.
Understanding Congruence
In our day-to-day life, we come across many objects which have the same shape
and same size. For example.
(i)
(ii)
Postal stamps of same denominations
Shaving blades of same brand of a company
3
(iii) Sheets of same letter paid
(iv) Coins of same denominations
4
(v)
(vi)
Toys made of the same mould
Biscuits in the same packet
Such objects are called congruent objects. The relation of two or more objects
being congruent is called congruence.
5
You can check that if you take one postal stamp as in (i) and place it over the other,
you will find that one covers the other exactly. Similarly, if you take one coin as in
(iv) and place it over the other, they cover each other exactly. Same is the case with
blades as in (ii) above, paper sheets as in (iii) above etc.
This idea of placing one object over the other and checking their congruence can
easily be used for congruence of any two plane figures. This process is called
superimposition or superposition.
2.
Congruence of Plane Figures
Look at the two figures given below:
Fig. 1
Can we say that these figures are congruent?
To check it, make a trace copy (or cut out) of any one of them and place it over the
other.
If they cover each other exactly, than you can say that they are congruent.
In this case, you will find that figure F1 exactly covers figure F2 and hence they are
congruent.
Symbol ‘ ’ is used to denote congruency of two figures. Thus,
F1
F2 or F2
F1,
and read as figure F1 is congruent to figure F2.
6
Now, check whether following figures F1 and F2 are congruent.
You will find they do not cover each other completely (exactly).
Hence, F1 is not congruent ( ) to F2
Example 1: Given below are some pairs of figures. By the method of super–
position, check which of them are congruent.
7
Fig. 2
Solution:
Figures F1 and F2 are congruent in (i), (ii), (iii), (v), (viii) and (ix).
Measure the line segments in figure (viii) above. Are the two length same? Yes,
they are.
So, we can say that
Two line segments are congruent if they have the same length.
Similarly, by measuring angles in figure (ix) above, you find that their measures
are same.
So, we can say that
Two angles are congruent if they have the same measure.
Similarly, by measuring the radii of the circles in figure (v) above, we can say that
Two circles are congruent if they have the same radius.
8
3.
Congruence of Triangles
Let ABC and DEF be two triangles (see Fig.3).
Fig. 3
To examine whether
ABC is congruent to
DEF, make a trace copy of the
triangles and check whether the two triangles cover each other exactly or not, by
using the method of superposition.
If they cover each other exactly, then obviously the two triangles are congruent
to each other.
In this case, you will observe that they cover each other exactly when
and
A falls on D
A
D
B falls on E
B
E
C falls on F
C
F
We express this fact as
ABC
Fig. 4
DEF
under the correspondence
and
A
D
B
E
C
F
9
This can also be expressed as
A
D
B
E
C
F
Fig. 5
Under this correspondence, you can easily see that
and
AB=DE,
A = D,
BC=EF
B= E
and
and
AC=DF
C= E
Thus, the two triangles are congruent if all the six parts (elements) of one
triangles are respectively equal to the six parts of the other triangle.
Let us try to superpose the trace copy of ΔABC and that of ΔDEF such that B falls
on D, and BC falls along DF.
Fig. 6
10
In this case, the two triangles do not cover each other exactly (Fig. 6)
Therefore, we cannot say that ΔABC is congruent to ΔDEF under the
correspondence.
B
D, C
F, A
E
So, we cannot write:
Δ ABC
Δ EDF
Similarly, two triangles may not be congruent in other correspondence also.
Example 2: If Δ PQR
(i)
P=
Δ CAB, then which one of the following is true?
C
(ii) PQ = AB
(iii) PR = CA
(iv)
R=
B
(v)
Q=
A
(vi) RQ = BC
Solution: Here correct correspondence is P
C, Q
A, and R
B.
So, (i) True, (ii) False, as PQ = CA (iii) False as PR = CB (iv) True (v) True (vi)
False, as RQ = BA
Example 3: If Δ ABC
(i)
Δ BCA
Δ EFD
(ii)
Δ ACB
Δ DFE
(iii) Δ CAB
Δ FED
Δ DEF, then which of the following is not correct?
11
(iv) Δ BAC
Δ EDF
Solution: If Δ ABC
B
E and C
Δ DEF, then correct correspondence is A
D,
F.
So, (i) correct (ii) correct (iii) not correct (iv) correct
4.
Criteria for Congruence of Triangles
You have seen that two triangles are congruent if all their six corresponding
parts, namely, three sides and three angles, are equal. But to verify or check the
congruence of two triangles, it is not always necessary to check the equality of all
these six parts. You will see that we can check the congruence of two triangles
even by checking equality of three specific corresponding parts instead of six
parts.
We now discuss these cases one by one.
Side-Side-Side (SSS) Criterion
Let us construct a triangle ABC with sides AB = 4 cm, BC = 5 cm and CA = 6
cm.
Construction
(i)
We first draw a rough figure [Fig. 7 (i)]
12
(ii)
Draw a line segment BC = 5 cm (you may take any side) first
[Fig. 7 (ii)].
..
(iii) With B as centre and 4 cm radius draw as arc on one side of BC [Fig. 7
(iii)].
(iv) With C as centre and 6 cm radius, draw an arc on the same side of BC
intersecting the arc in step (iii) at A [Fig. 7 (iv)].
13
(v)
Join AB and AC [Fig. 7 (v)].
Then, Δ ABC is the required triangle
Now construct one more triangle DEF with same measurement of sides i.e.
DE = 4 cm, EF = 5 cm and DF = 6 cm [Fig. 7 (vi)]
Fig. 7
Activity-1
Make a trace copy of either of these two triangles (Δ ABC or Δ DEF) and place one
triangle over the other and see whether they cover each other exactly in some position.
You will find that they are covering each other when A falls on D, B falls on E and C
falls on F.
Thus,
14
Δ ABC
Δ DEF (by superposition).
Activity 2:
Measure
A,
B,
C and
You will also find that
D,
A=
E,
D,
F.
B=
E and
C=
F. i.e., all the six corresponding
parts of the two triangles are equal. Hence, the two triangles are congruent.
From Activity 1 and Activity 2, we can say that
Two triangles are congruent if three sides of one triangle are respectively equal to
three sides of the other triangle.
This is known as side-side-side – (SSS) Criterion (or rule or property) for congruency
of two triangles.
•
Side-Angle-Side Criterion
Let us construct a triangle ABC, where BC = 5 cm, AB = 4 cm and
B = 60º.
Construction:
(i)
First draw a rough figure [Fig. 8 (i)] showing the given measurements.
(ii)
Draw a line segment BC = 5 cm [Fig. 8 (ii)].
15
(iii)
Construct an angle XBC of 60º at the point B [Fig. 8 (ii)].
(iv)
With B as centre and 4 cm radius, draw an arc intersecting the ray
BX at A [Fig. 8 (iv)].
(v)
Join AC. [Fig. 8 (v)]
Δ ABC is the required triangle.
Now construct one more triangle DEF with same measurements, i.e., EF = 5 cm,
60º, DE = 4 cm [Fig. 8(vi)].
16
E=
Fig. 8
Activity 3:
Make a trace copy of either of these two triangles (Δ ABC or Δ DEF) and place one over
the other and see whether they cover each other exactly in some position. You will find
that they are covering each other when A falls on D, B falls on E and C falls on F. That
is, they cover each other under the correspondence
A
D, B
E, C
Thus, Δ ABC
F.
Δ DEF (by superposition).
Activity 4:
Measure
A,
C, side AC;
D,
F, side DF
i.e., all the six corresponding parts of the two triangles are equal. Hence, the two
triangles are congruent.
From Activity 3 and Activity 4, we can say that
Two triangles are congruent if two sides and the included angle of one triangle are
respectively equal to two sides and the included angle of the other triangle.
This is known as Side-Angle-Side (SAS) Criterion for congruency of two triangles.
•
Angle-Side-Angle Criterion
Let us construct a triangle ABC, where BC = 5.5 cm,
17
B = 30º and
C = 100º
Construction
(i)
First draw a rough figure showing the given measurements [Fig. 9(i)].
(ii)
Draw a line segment BC = 5.5 cm [Fig. 9(ii)].
(iii)
Construct an angle XBC = 30º at the point B [Fig. 9(iii)].
(iv)
Draw an angle YCB = 100º at the point C.
Let ray CY interest ray BX at the point A. Join AC [Fig. 9(iv)].
18
Δ ABC is the required triangle.
Now, construct one more triangle DEF with same measurements, i.e., EF = 5.5 cm,
30º and
E=
F = 100º.
Activity 5 :
Make a trace copy of either of these two triangles ( ABC or ( DEF) and place one over
the other and see whether they cover each other exactly in some position.
You will find that they are covering each other when A falls on D, B falls on E and C
falls on F. That is, they cover each other under the correspondence
A
D, B
Thus, Δ ABC
E, C
F.
Δ DEF (by superposition).
Fig. 9
19
Activity 6:
Measure
A, AB and AC of Δ ABC and
D, DE and DF of Δ DEF.
You will also find that
A=
D, AB = DE and AC = DF.
i.e., all the six corresponding parts of the two triangles are equal. Hence, the two
triangles are congruent.
From Activity 5 and Activity 6, we can say that
Two triangles are congruent if two angles and the included side of one triangle are
respectively equal to two angles and the included side of the other triangle.
This is known as Angle-Side-Angle (ASA) criterion for congruency of two triangles.
Note that if two angles of one triangle are respectively equal to two angles of other
triangle, then by angle sum property, the 3rd angles of the two triangles with also be
equal. In view of the above, we can say that
If two angles and a side of one triangle are respectively equal to two angles and the
corresponding side of the other triangle, then the two triangles are congruent.
This is known as Angle-Angle-Side (AAS) Criterion for congruence of two triangles.
For example, if in
Δ ABC
But if
ABC and
DEF, if
A =
D,
B =
E and BC = EF, then
Δ DEF by AAS.
A=
D,
B=
E and BC = DF, then the triangles may or may not be congruent.
20
Δ ABC
Δ DEF
Δ ABC may not be congruent to Δ DEF
Right-Hypotenuse-Side (RHS) Criterion
Let us construct a Δ ABC in which
B = 90º, hypotenuse AC = 8 cm and side AB = 4.5
cm.
Construction:
(i)
First draw a rough figure [Fig. 10(i)], showing given measurements.
(ii)
Draw a line segment AB = 4.5 cm [Fig. 10(ii)].
21
(iii)
Construct an angle XBA of 90º at the point B [Fig. 10(iii)].
X
(iv)
With A as centre and radius 8 cm, draw an arc intersecting ray BX at the point
C. [Fig. 10(iv)]. Join AC.
Δ ABC is the required triangle.
Now construct one more triangle DEF with the same measurements, i.e.,
8 cm and DE = 4.5 cm. [Fig. 10(v)].
22
E = 90º, DF =
(v)
Fig. 10
Activity 7:
Make a trace copy of either of these two triangles (Δ ABC or Δ DEF) and place one
triangle over the other and see whether they cover each other exactly in some position.
You will find that they are covering each other when A falls on D, B falls on E and C
falls on F. Thus.
Δ ABC
Δ DEF (by superposition).
(vi)
Fig. 10
23
Activity 8:
Measure
A,
find that
A=
C and side BC of ΔABC and
D,
C=
D,
F and side EF of
DEF. You will also
F and BC = EF, i.e., all the six corresponding parts of the two
triangles are equal.
Hence, the two triangles are congruent.
From Activity 7 and Activity 8, we can say that
Two right triangles are congruent if hypotenuse and one side of one triangle are
respectively equal to the hypotenuse and one side of the other triangle.
This is known as Right-Hypotenuse-Side (RHS) criterion for congruency of two
triangles.
We now explain the use of these congruent criteria in solving problems.
Example 4: In Δ ABC and Δ PQR,
AB = 4.5 cm, AC = 6 cm, PQ = 4.5 cm and PR = 6 cm.
What additional information will you need to establish the congruency of two triangles
using
(i)
SSS Criterion?
(ii)
SAS Criterion?
Solution:
(i)
Additional information required is C third side of two triangles must be
equal, i.e., BC = QR
(ii)
A=
P. (included angles)
24
Example 5:
In each of the following pair of triangles, identify whether they are congruent or not. If
congruent, state the congruent criterion and also write the triangles in symbolic form :
(i)
(ii)
(iii)
(iv)
25
(v)
(vi)
(vii)
(viii)
26
(ix)
Solution :
(i)
Congruent; SAS; Δ ABC
(ii)
Not congruent
Δ PQR
(iii) Not congruent
(iv) Congruent; SSS; Δ ABC
(v)
Δ FDE
Not congruent
(vi) Congruent, RHS; Δ KIT
Δ NLM
(vii) Congruent; ASA; Δ PQR
Δ ABC
(viii) Not congruent
(ix) Congruent; SAS; Δ ABC
Δ RQP
Example 6: In the following figure.
AB = AD,
BAC =
Is Δ ABC
Δ ADC?
CAD.
Give reason.
Solution:
(i)
Yes, Δ ABC
Δ ADC
as AB = AD (Given)
BAC =
DAC (Given)
27
AC = AC (Common)
So, triangles are congruent by SAS Criterion.
Example 7: In the figure, D is the mid–point of side BC of Δ ABC and AB = AC.
(i)
Is Δ ABD
(ii)
Is
B=
Δ ACD? Give reason.
C? Give reason.
Solution:
As AB = AC (Given)
AD = AD (Common)
BC = CD (As D is the mid–point of BC)
So, Δ ABD
Δ ACD by SSS criterion
Thus,
C [Corresponding parts of congruent triangles (CPCT)]
B=
Check that Δ ABC is an isosceles triangle.
So, we can say that angles opposite to equal sides are
also equal.
Examples 8:
In the figure, R is the mid–point of the line segment PT.
Find
SRT and
Is Δ PQR
QPR.
Δ TSR?
Is QR = SR?
Solution:
QPR = 180º - (105º + 50º)
28
[Angle Sum Property]
= 180º - 155º
= 25º
SRT =
PRQ = 50º (vertically opposite angles)
AS PR = TR (R is the mi–point of PT)
P=
T
PRQ =
TRS
So, Δ PQR
Δ TSR (by ASA)
Thus, QR = SR (CPCT)
Example 9: In the figure, in Δ PQR, altitude QS = altitude RT.
Is Δ QRT
Δ RQS?
Is
PRQ?
PQR =
Solution:
In Δ QRT and Δ RQS,
QTR =
RSQ = 90º
Hypotenuse QR = Hypotenuse PQ (Common)
TR = SQ (Given)
So, Δ QRT
Yes,
PQR =
Δ RQS (RHS)
PRQ (CPCT)
29
Example 10: In the figure, line segments AE and BD bisect each other at the
(i)
Is Δ ABC
Δ CDE? Give reason
(ii)
Is Δ ABC
Δ ECD? Give reason
(iii)
Is
A=
D? Give reason
(iv)
Is
A=
E? Give reason
(v)
Is AB = DE? Give reason
(vi)
Is AB║DE? Give reason
Solution:
(i)
No, as BC
DE.
(ii)
Yes, AC = EC (AE and BD bisect each other)
BC = DC
ACB =
ECD (Vert. Opp. Angles)
30
point C.
So, triangles are congruent by SAS
(iii)
No, there are not corresponding parts of congruent triangles
(iv)
Yes, CPCT
(v)
Yes, CPCT
(vi)
Yes, as
A=
E and they are alternate interior angles.
Example 11: Construct a line parallel to a given line ℓ from a point P not lying ℓ.
Solution: Let ℓ be a line and P be a point not lying on the line ℓ.
We have to construct a line through P, parallel to ℓ, using ruler and a compass.
Steps:
1.
Take any point A on ℓ.
2.
Join PA
31
3.
With A as centre and a convenient radius, draw an arc to intersect line ℓ at B and
line segment AP at C.
4.
With P as centre and the same radius as in Step 3, draw an arc to intersect PA at D.
5.
With D
as centre and radius equal to BC, draw an arc cutting the arc of step 4 at E as
shown in Fig. (vi).
P
6.
Join PE and extend it to form a line m.
Line m is the required line through point P parallel to line ℓ.
Justification for the Construction
Here
CAB =
DPE [Δ ABC
Δ PDE (SSS) and
But they are alternate interior angles
So, m║ ℓ.
32
A=
P (CPCT)]
Student’s
Support
Material
33
Student’s Worksheet - 1
Shapes
Warm Up (W 1)
Name of the student ______________________
Date ______________
Activity – Shapes
Write 5 real life objects having the given shape, in the table below
Shape
Real life object
Extension Activity: Make a Mosaic using various shapes, mentioned above.
34
Student’s Worksheet - 2
Tessellation
Warm Up (W 2)
Name of the student ______________________
Date ______________
Activity –Tessellation maker
A tessellation is the name given to a type of pattern made up of shapes
of same size which interlock without overlapping or leaving any gaps.
A)
Follow the links given below to make some amazing online tessellations.
http://illuminations.nctm.org/ActivityDetail.aspx?ID=205http://illuminations.nc
tm.org/ActivityDetail.aspx?ID=202
B)
Find 3 examples of tessellations in nature as well as man made tessellations.
Tessellations in nature:
Man made tessellations:
C)
Create your own tessellation masterpiece on the given graph paper.
35
36
Student’s Worksheet - 3
Congruence
Pre Content Worksheet (P1)
Name of the student ______________________
Date ______________
Activity –Size Sorting
A)
Observe and find which of the following pairs of pictures are of same size.
Colour the pictures having same size.
37
Discuss with your friend about the pictures which are not of same size. What do
they have common among them?
________________________________________________________________________
________________________________________________________________________
Follow the given link to understand the concepts of Congruence and Similar
shapes.
http://www.harcourtschool.com/activity/similar_congruent/
Discuss with your friend and write definition of Congruence.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Discuss and write definition of Similar shapes.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
B)
List 5 examples of real life objects having congruence.
e.g. Playing cards in a deck of cards.
1.
2.
3.
4.
5.
C)
Tracing and Superimposition
Follow the given link and practice congruence by superimposition using online
manipulative.
http://www.learner.org/courses/teachingmath/grades3_5/session_02/section
_02_b.html
38
Measure and find which of the figures in the given grids are congruent? Trace figures
from second grid to use method of superimposition.
Discuss with your friend and find an alternate method to find congruent figures in the
grids given above.
______________________________________________________________________________
______________________________________________________________________________
______________________________________________________________________________
39
Student’s Worksheet - 4
Congruent Figures
Content Worksheet (CW 1)
Name of the student ______________________
Date ______________
Activity – Net Size
A)
Follow the given link and make nets of cube, cuboids and tetrahedron.
Step 1: Select a shape, rotate and observe the various faces. Discuss with
your friend if the faces of the selected solid are congruent?
Step 2: Make net by doing the appropriate selection. Find the number of
congruent figures.
http://illuminations.nctm.org/ActivityDetail.aspx?ID=70
B)
Observe the given nets of solid shapes and colour the congruent faces in same
colour. Make the solid shapes thus formed in front of the net.
40
Are faces of solid shapes congruent? Explain.
________________________________________________________________________
________________________________________________________________________
________________________________________________________________________
Solids having all faces congruent are _______________________________________
Extension Activity
1.
Game
Follow the given link, play the game and appreciate congruent figures.
http://www.fuelthebrain.com/Game/play.php?ID=131
2.
Online Congruent Figures Quiz
Follow the given link and test your understanding of the concept.
http://www.mathopolis.com/questions/q.php?id=1770&site=1&ref=/geometry/
congruent.html&qs=1770_779_780_1769_3270_3271
41
Student’s Worksheet -5
Congruence of Triangles(SSS)
Content Worksheet (CW 2)
Name of the student ______________________
Date ______________
Activity- SSS Congruence
A)
Follow the given link to understand SSS Congruence Criteria
http://www.mathopenref.com/congruenttriangles.html
1.
What do you understand by SSS Congruence Criteria for triangles?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
2.
How many measures do you need to know, to prove triangles congruent?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
3.
What do you mean by symbolic form of congruent triangles?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
4.
B)
What do you understand by corresponding parts of congruent triangles?
__________________________________________________________________
__________________________________________________________________
__________________________________________________________________
In the given grid draw the triangles congruent to the given triangles, such that
1. ALM
POT
2.
PSR
WIN
42
Complete the blanks:
C)
A = ___________
L = ___________
M = __________
P = ___________
S = ___________
R = __________
Write the matching partsand the symbolic form for each part, ifgiven trianglesare
congruent.
1.
ABC is isosceles and D is the mid-point of AC.
2. Here, CA = AD and BC = BD. Is C = D? Give reason.
43
3. AD =5cm, EB =5cm, C is the mid-point of AB and ED.
Write true or false:
a)
ADC
ECB.
b)
ADC
CBA.
c)
ECB
d)
A = B.
DCA.
4. Prove congruence. Is B = 900? Give reason.
5. Prove congruence and find if P = R or not. Give reason.
44
Extension Activity: Find how many triangles in the given figure are congruent to ABE
by taking trace copy of ABE and cutting it out.
Student’s Worksheet – 6
Construction of triangle (SSS)
Content Worksheet (CW 3)
Name of the student ______________________
Date ______________
Activity – SSS construction
1.
Construct TOD such that TO = 5.5 cm, OD = 6.8cm and TD = 9cm.
2.
Construct LNM in which LN = 5cm and LM = MN = 9.5cm.What type of triangle
is this?
45
3.
Construct TAP in which TA = 5cm, AP = 3cm and TP = 4cm. What type of
triangle is this? Give reasons.
4.
Construct 
PQR in which PQ=5.3cm, PR=4.6cm and QR=9.8cm.What type of
triangle is this?
5.
Is it possible to construct a triangle whose sides are 6.5cm, 9 cm and 2.5? Explain.
Student’s Worksheet –7
Congruence of Triangles(SAS)
Content Worksheet (CW 4)
Name of the student ______________________
Date ______________
Activity - SAS Forum
1.
What do you understand by SAS Congruence Criteria for triangles?
2.
Explain the term ‘included angle’.
46
3.
If AC = AD and
CAB =
DAB. Prove the congruence of the given triangles.
Is CB = DB.
4.
Write the matching partsand the symbolic form and check congruence of ABQ
and ABP.
5.
In AFN, FO is the perpendicular bisector of AN.
a) Write the matching parts to prove congruence of the given triangles.
b) Write the symbolic form and the congruence condition applied.
c) Is A = N
6.
Name the additional equal corresponding part(s) needed to prove the congruence
of triangles by SAS condition.
47
7.
i)
ii)
iii)
Student’s Worksheet –8
Construction of Triangle (SAS)
Content Worksheet (CW 5)
Name of the student ______________________
Date ______________
Activity - SAS construction
1.
Construct PQR in which QR=5.2cm,
Q=120°
and PQ=3.5cm. What type of triangle is this?
2.
Construct an isosceles right triangle whose each of equal sides is of 6.5cm. Label it.
48
3.
In ABC, AB = 12, BC = 12, and
B = 45°. In XYZ, XY = 12, YZ = 12, and
X= 45°.
Are the triangles congruent?
4.
Construct PQR in which PQ = 5cm,
Q = 300 and QR = 5cm. Measure
R.What
type of triangle is this?
5.
Construct a right angled triangle with sides containing the right angles as 12cm
and 5cm. Measure the hypotenuse.
Student’s Worksheet –9
Congruence of Triangles (ASA)
Content Worksheet (CW 6)
Name of the student ______________________
Date ______________
Activity - ASA Forum
1.
What do you understand by ASA Congruence Criteria for triangles?
49
2.
Explain the term ‘included side’.
3.
If AB = XY,
CBA =
ZYX and
CAB =
ZXY. Prove the congruence of the given
triangles. Is CB = ZY.
4.
In the figure which two triangles are congruent to each other by ASA congruence?
Give reason.
5.
State what additional information is required in order to know that the triangles
are congruent by ASA congruence.
50
6.
In the given figure by applying ASA congruence rule, start whether the triangles
are congruent. If yes, write them in symbolic form.
7.
Ray AX bisects
DAB as well as
DCB.
(a)
State the three pairs of equal parts in ∆BAC and ∆DAC
(b)
Is ∆BAC congruent to ∆DAC? Give reasons.
(c)
Is AB = AD? Justify your answer.
(d)
Is CD = CB? Give reasons.
51
Student’s Worksheet –10
Construction of Triangles (ASA)
Content Worksheet (CW 7)
Name of the student ______________________
Date ______________
Activity - ASA construction
X = 60o,
1.
Construct a ∆XYZ given
2.
Construct ∆ABC if AB = 5cm,
3.
Construct a triangle PQR in which
Y = 30o and XY = 5.8cm.
B =105o and
Q=
C = 40o.
R = 60o and PQ = 5.4cm? Can you find
PR? Name the triangle according to its sides.
52
4. Construct ∆DEF with EF = 3.8cm,
E = 60o and
F = 30o. Measure
D. What
type of triangle is this?
5. Can you construct ∆LMN such that MN = 7.3cm,
M = 110o and
N = 70o? Give
reason in support of your answer.
Student’s Worksheet –11
Congruence of Triangles (RHS)
Content Worksheet (CW 8)
Name of the student ______________________
Date ______________
Activity - RHS Forum
1.
In the given figure AB = DC, AB is perpendicular to AD and DC is perpendicular
to BC. Is AD = BC? Give reasons.
53
2.
In ∆ABC and ∆FEC BC = DE. State the three reasons to make the triangles
congruent. Which rule did you use? Is AD = BC?
3.
In ∆PQR, PS is perpendicular to QR and PQ = PR. Prove that ∆PQS
∆PRS.Is PS a
perpendicular bisector of QR?
4.
What additional information do you need to prove that the triangles are congruent
by RHS congruence rule? State what is already known.
5.
In the given figure ABCD is a parallelogram. Prove which two triangles are
congruent by RHS congruence rule.
54
Student’s Worksheet –12
Construction of Triangle (RHS)
Content Worksheet (CW 9)
Name of the student ______________________
Date ______________
Activity - RHS construction
1.
Construct a right angled triangle whose hypotenuse is
6cm long and one of the other two sides is 4cm long.
2.
Construct a right angled triangle in which sides containing the right angle is 3cm
and 4cm. Measure the hypotenuse.
3.
Draw a right angled triangle having hypotenuse of length 6.5cm and one of the
acute angles be 45o. Name the type of triangle formed.
4.
Construct a right angled ∆XYZ right angled at Y, YZ = 8cm and XZ = 10cm.
55
Student’s Worksheet –13
Construction of Parallel Lines
Content Worksheet (CW 10)
Name of the student ______________________
Date ______________
Activity - Parallel lines construction
Follow the steps and construct a line parallel to a given line from a point outside it.
(Two steps are shown)
STEPS
YOUR WORK
1. Start with a line PQ and a
point R off the line.
2. Draw a transverse Line
through R and across the line
PQ at an angle, forming a
point X where it intersects the
line PQ. The exact angle is not
mandatory.
3. With the compass width set to
about
half
the
distance
between R and J, place the
point onJ, and draw an arc
across both lines.
56
4. Without
adjusting
the
compass width, move the
compass to R and draw a
similar arc to the one in step 2
on the other side of the
transversal.
5. Set compass width to the
distance where the lower arc
crosses the two lines.
6. Move the compass to where
the upper arc crosses the
transverse line and draw an
arc across the upper arc,
forming point S.
7. Draw a straight line through
points R and S.
8. Done. The line RS is parallel
to the line PQ
57
Student’s Worksheet –14
Skill Drill 1
Post-Content Worksheet (PCW 1)
Name of the student ______________________
Date ______________
Activity - More practice
1.
Lengths of the sides of triangle are indicated. By applying SSS
congruence rule, state which parts of triangles are congruent. In case of
congruent triangles write the result in symbolic form:
a)
b)
c)
58
2.
State the congruent triangles in the figure given.
3.
Triangle ABC and triangle ABD are on a common base AB and AC=BD and
BC=AD as shown in figure.
4.
Let the correspondence YZX
each statement correct.
RAP be true. Then fill in the blanks to make
a) Y
b)XZ
c) X
d)XY
e)ZY
f)
5.
In the figure E is the midpoint of A and CD, prove that ∆ACE=∆BDE.
59
6.
By applying SAS congruence rule, state the pairs of congruent triangles in the
following figure, if there is any, write them in symbolic form
a)
7.
In the figure AB=AD and
=
find the third pair of corresponding parts
to make ∆ABC ∆ADC by SAS congruent condition.
60
8.
In the adjacent figure AB=RS, RP=BQ and
=
prove that ∆ABP=∆SRQ
9.
a)
Is ∆ZYX ZPQ
b)
If yes write three pairs of matching parts you have used to arrive at the
conclusion.
10. in ∆ABC, AD is the angle bisector of
that ∆ABC is an isosceles triangle.
11.
such that AD is perpendicular to BC. Prove
BD and CE are altitudes of ∆ABC, such that BD=CE.
a) Is ∆CBD ∆BCE? Why or why not?
b) Is ∆
∆BCE? Why or why not?
61
12.
BD and CE are altitudes of ∆ABC, such that BD=CE.}
a)
State three pairs of matching parts in ∆CBD and ∆BCE
b)
Is
. Why or why not?
13.
In the given figure, DE and DF are the perpendiculars drawn on BA and BC
respectively, such that DE=DF, prove that ∆DEB ∆DFB
14.
ABC is a triangle with D as mid-point of AB. DE is perpendicular to AC and DF
is perpendicular to BC. Also DE=DF. Prove that ∆ABC is an isosceles triangle.
62
Student’s Worksheet –15
Skill Drill -2
Post-Content Worksheet (PCW 2)
Name of the student ______________________
Date ______________
Activity - Independent practice
1.
2.
3.
If two figures have equal area then they are
(a)
Congruent
(c)
May or may not be congruent
(a)
2 right angles
(b)
2complete angles
(c)
2staright angles
(d)
2zero angles
b)
False
All squares are congruent
True
b)
False
Which of the following is not a congruence rule
(a)
6.
True
If the hypotenuse of one right triangle is equal to the hypotenuse is equal to
hypotenuse of another triangle then the triangle are congruent
a)
5.
Not congruent
The sum of angles of a quadrilateral is
a)
4.
(b)
AAS
(b)
ASA
(c)
SAS
(d)
AAA
SSA
(d)
SSS
Which of the following is not a congruence rule
(a)
SAS
(b)
SSS
(c)
63
7.
Show that ABC
State the condition used:
8.
In the adjoining figure PA
ABQ. Is BP=AQ?
9.
Show that ABC
10.
A
D
B
C
AB and QB
AB, also AP=QB . Prove that BAP
EFD? State matching pairs:
Draw rhombus ABCD.Are ADC and ABC congruent? What can you say about
ABD and BCD?
64
11.
If PQR
LMN and PQ=6cm,PR=5cm and P=500 then find NL, LM and L.
Q.12- Q.14 In the following figures, name the triangle, which is congruent to  ABC,
giving the right correspondence? Give reason.
12
13.
14.
65
15.
Draw a line ‘m’, Draw a perpendicular to ‘m’ at any point on ‘m’. On this
perpendicular choose a point X which is 4cm away from ‘m’. Through X draw a
line ‘n’ parallel to ‘n’.
16.
Construct an equilateral triangle each of whose sides are of length 5.4cm.
17.
Construct a right triangle DEF such that DE = 5cm, E = 90o and hypotenuse
DF = 13cm. Hence find EF.
18.
Construct a ∆ABC with AB = AC in which BC = 6.4cm and base angle is 30o.
19.
Construct a ∆PQR such that PQ = 6cm, Q = 100o, P = 40o. Measure
20.
Construct a ∆ABC such that BC = 5cm, BC = 7cm and BA = 5.2cm.
66
Suggested Video Links
Name
Title/Link
Video Clip 1
www.worldofteaching.com/powerpoints/.../geometrycongruence.ppt
- Similar
http://www.google.co.in/search?aq=0&oq=ppt+on+congruenc
Video Clip 2
&ix=seb
Video Clip 3
Video Clip 4
www.ih.k12.oh.us/MSDunlap/Geometry/Congruent%20Triangles.p
pt - Similar
http://www.authorstream.com/Presentation/aSGuest52368429792-cong-triangles-3-education-ppt-powerpoint/
Video Clip 5
educator.schools.officelive.com/Documents/ssssasasaaas5.55.6.ppt
Video Clip 6
http://www.mathwarehouse.com/geometry/congruent_triangl
es/
Video Clip 7
paccadult.lbpsb.qc.ca/eng/extra/img/Congruent%20Triangles.ppt Similar
Video Clip 8
www.taosschools.org/ths/Departments/MathDept/quintana/
GeometryPPTs/4.4%20ASA%20AND%20AAS.ppt
Weblink1
http://www.learner.org/courses/teachingmath/grades3_5/ses
sion_02/section_02_b.html
Weblink 2
http://www.beaconlearningcenter.com/WebLessons/Congrue
ntConcentration/concentration.htm
Weblink 3
http://www.harcourtschool.com/activity/similar_congruent/
Weblink 4
http://www.youtube.com/watch?v=pZZADeU3khw&feature=
related
Weblink 5
www.youtube.com/watch?v=dvwdwslCyWA
Weblink 6
http://www.youtube.com/watch?v=P8Fnaw1Sxtw
Weblink 7
www.youtube.com/watch?v=NAhcmPS5k9g
67
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