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CE CONGRUENCE CONGRUENCE CONGRUENCE CLASS CO GRUENCE CONGRUENCE CONGRUENCE CONGRUEN CE CONGRUENCE CONGRUENCE CONGRUENCE CO GRUENCE CONGRUENCE CONGRUENCE CONGRUEN CE CBSE-i CONGRUENCE CONGRUENCE CONGRUENCE CO UNIT-10 GRUENCE CONGRUENCE CONGRUENCE CONGRUEN CE CONGRUENCE CONGRUENCE CONGRUENCE CO GRUENCE CONGRUENCE CONGRUENCE CONGRUEN 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 The CBSE-International is grateful for permission to reproduce and/or translate copyright material used in this publication. The acknowledgements have been included wherever appropriate and sources from where the material may be taken are duly mentioned. In case any thing has been missed out, the Board will be pleased to rectify the error at the earliest possible opportunity. All Rights of these documents are reserved. No part of this publication may be reproduced, printed or transmitted in any form without the prior permission of the CBSE-i. This material is meant for the use of schools who are a part of the CBSE-International only. This International Curriculum initiated by Central Board of Secondary Education - (CBSE) is a progressive step in making the educational content and methodology more sensitive and responsive to the global needs. It signifies the emergence of a fresh thought process in imparting a curriculum which would restore the autonomy of the learner to pursue the learning process in harmony with the existing personal, social and cultural ethos. The Central Board of Secondary Education has been providing support to the academic needs of the learners worldwide. It has about 12500 schools affiliated to it and over 158 schools situated in more than 23 countries. The Board has always been conscious of the varying needs of the learners and has been working towards contextualizing certain elements of the learning process to the physical, geographical, social and cultural environment in which they are engaged. The International Curriculum being designed by CBSE-i, has been visualized and developed with these requirements in view. The nucleus of the entire process of constructing the curricular structure is the learner. The objective of the curriculum is to nurture learner autonomy, given the fact that every learner is unique. The learner has to understand, appreciate, protect and build on values, beliefs and traditional wisdom, make the necessary modifications, improvisations and additions wherever and whenever necessary. The recent scientific and technological advances have thrown open the gateways of knowledge at an astonishing pace. The speed and methods of assimilating knowledge have put forth many challenges to educators, forcing them to rethink their approaches for knowledge processing by their learners. In this context, it has become imperative for them to incorporate those skills which will enable young learners to become 'life long learners'. The ability to stay current, to upgrade skills with emerging technologies, to understand the nuances involved in change management and the relevant life skills have to be a part of the learning domains of the global learners. The CBSE-i curriculum has taken cognizance of these requirements. The CBSE-i aims to carry forward the basic strength of the Indian system of education while promoting critical and creative thinking skills, effective communication skills, interpersonal and collaborative skills along with information and media skills. There is an inbuilt flexibility in the curriculum, as it provides a foundation and an extension curriculum, in all subject areas to cater to the different pace of learners. The CBSE introduced classes I and X in the session 2010-11 as a pilot project in schools. It was further extended to classes II, VI and X in the session 2011-12. In the seesion 2012-13, CBSE-i is going to enter in third year with classes III, VII and XI. The focus of CBSE-i is to ensure that the learner is stress-free and committed to active learning. The learner would be evaluated on a continuous and comprehensive basis consequent to the mutual interactions between the teacher and the learner. There are some non-evaluative components in the curriculum which would be commented upon by the teachers and the school. The objective of this part or the core of the curriculum is to scaffold the learning experiences and to relate tacit knowledge with formal knowledge. This would involve trans-disciplinary linkages that would form the core of the learning process. Perspectives, SEWA (Social Empowerment through Work and Action), Life Skills and Research would be the constituents of this 'Core'. The Core skills are the most significant aspects of a learner's holistic growth and learning curve. The International Curriculum has been designed keeping in view the foundations of the National Curricular Framework (NCF 2005) NCERT and the experience gathered by the Board over the last seven decades in imparting effective learning to millions of learners, many of whom are now global citizens. The Board does not interpret this development as an alternative to other curricula existing at the international level, but as an exercise in providing the much needed Indian leadership for global education at the school level. The International Curriculum would evolve building on learning experiences inside the classroom over a period of time. The Board while addressing the issues of empowerment with the help of the schools' administering this system strongly recommends that practicing teachers become skillful learners on their own and also transfer their learning experiences to their peers through the interactive platforms provided by the Board. I profusely thank Shri G. Balasubramanian, former Director (Academics), CBSE, Dr. Sadhana Parashar, Director (Training) CBSE, Dr. Srijata Das, Education Officer CBSE, CBSE along with all the Officers involved in the development and implementation of this material. The CBSE-i website enables all stakeholders to participate in this initiative through the discussion forums provided on the portal. Any further suggestions for modifying any part of this document are welcome. Vineet Joshi Chairman , CBSE Advisory Conceptual Framework Shri Vineet Joshi, Chairman, CBSE Dr. Sadhana Parashar, Director (Training), Shri G. Balasubramanian, Former Director (Acad), CBSE Ms. Abha Adams, Consultant, Step Dr. Sadhana Parashar, Director (Training) Ideators VI-VIII Ms Aditi Mishra Ms Guneet Ohri Ms. Sudha Ravi Ms. Himani Asija Ms. Neerada Suresh Ms Preeti Hans Ms Neelima Sharma Ms. Gayatri Khanna Ms. Urmila Guliani Ms. Anuradha Joshi Ms. Charu Maini Dr. Usha Sharma Prof. Chand Kiran Saluja Dr. Meena Dhani Ms. Vijay Laxmi Raman Material Production Groups: Classes VI-VIII English : Physics : Mathematics : Ms. Vidhu Narayanan Ms Neha Sharma Ms. Deepa Gupta Ms. Meenambika Menon Ms Dipinder Kaur Ms. Gayatri Chowhan Ms. Patarlekha Sarkar Ms Sarita Ahuja Ms. N Vidya Ms. Neelam Malik Ms Gayatri Khanna Ms. Mamta Goyal Ms Preeti Hans Ms. Chhavi Raheja Biology: Ms Rachna Pandit Mr. Saroj Kumar Ms Renu Anand Hindi: Ms. Rashmi Ramsinghaney Ms Sheena Chhabra Mr. Akshay Kumar Dixit Ms. Prerna Kapoor Ms Veena Bhasin Ms. Veena Sharma Ms Trishya Mukherjee Ms. Seema Kapoor Ms. Nishi Dhanjal Mr. Manish Panwar Ms Neerada Suresh Ms. Kiran Soni Ms. Vikram Yadav Ms Sudha Ravi Ms. Monika Chopra Ms Ratna Lal Ms Ritu Badia Vashisth Ms. Jaspreet Kaur CORE-SEWA Ms Vijay Laxmi Raman Ms. Preeti Mittal Ms. Vandna Ms. Shipra Sarcar Ms.Nishtha Bharati Chemistry Ms. Leela Raghavan Ms.Seema Bhandari, Ms. Poonam Kumar Ms. Seema Chopra Mendiratta Ms. Madhuchhanda Ms. Rashmi Sharma MsReema Arora Ms. Kavita Kapoor Ms Neha Sharma Ms. Divya Arora Ms. Malini Sridhar Ms. Leela Raghavan Dr. Rashmi Sethi Ms. Seema Rawat Ms. Suman Nath Bhalla Geography: Ms Suparna Sharma Ms Aditi Babbar History : Ms Leeza Dutta Ms Kalpana Pant Ms Ruchi Mahajan Political Science: Ms Kanu Chopra Ms Shilpi Anand Economics : Ms. Leela Garewal Ms Anita Yadav CORE-Perspectives Ms. Madhuchhanda, RO(Innovation) Ms. Varsha Seth, Consultant Ms Neha Sharma Coordinators: Ms. Sugandh Sharma, E O Dr. Srijata Das, E O (Chief Co-ordinator, CBSE-i) Dr Rashmi Sethi, E O Ms.S. Radha Mahalakshmi, EO Mr. Navin Maini, R O (Tech) Ms. Madhu Chanda, R O (Inn) Shri Al Hilal Ahmed, AEO Mr. R P Singh, AEO Ms. Anjali, AEO Shri R. P. Sharma, Consultant (Science) Ms. Neelima Sharma, Consultant (English) Mr. Sanjay Sachdeva, S O 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 CE CONGRUENCE CONGRUENCE CONGRUENCE CO GRUENCE CONGRUENCE CONGRUENCE CONGRUEN CE CONGRUENCE CONGRUENCE CONGRUENCE CO GRUENCE CONGRUENCE CONGRUENCE CONGRUEN CE CONGRUENCE CONGRUENCE CONGRUENCE CO GRUENCE CONGRUENCE CONGRUENCE CONGRUEN CE CONGRUENCE CONGRUENCE CONGRUENCE CO GRUENCE CONGRUENCE CONGRUENCE CONGRUEN CENTRAL BOARD OF SECONDARY EDUCATION Shiksha Kendra, 2, Community Centre, Preet Vihar, Delhi-110 092 India