Page 1 CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a triangle. (â€˜Triâ€™ means â€˜threeâ€™). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices. In Chapter 6, you have also studied some properties of triangles. In this chapter, you will study in details about the congruence of triangles, rules of congruence, some more properties of triangles and inequalities in a triangle. You have already verified most of these properties in earlier classes. We will now prove some of them. 7.2 Congruence of Triangles You must have observed that two copies of your photographs of the same size are identical. Similarly, two bangles of the same size, two A TM cards issued by the same bank are identical. You may recall that on placing a one rupee coin on another minted in the same year, they cover each other completely. Do you remember what such figures are called? Indeed they are called congruent figures (â€˜congruentâ€™ means equal in all respects or figures whose shapes and sizes are both the same). Now, draw two circles of the same radius and place one on the other. What do you observe? They cover each other completely and we call them as congruent circles. Fig. 7.1 Page 2 CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a triangle. (â€˜Triâ€™ means â€˜threeâ€™). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices. In Chapter 6, you have also studied some properties of triangles. In this chapter, you will study in details about the congruence of triangles, rules of congruence, some more properties of triangles and inequalities in a triangle. You have already verified most of these properties in earlier classes. We will now prove some of them. 7.2 Congruence of Triangles You must have observed that two copies of your photographs of the same size are identical. Similarly, two bangles of the same size, two A TM cards issued by the same bank are identical. You may recall that on placing a one rupee coin on another minted in the same year, they cover each other completely. Do you remember what such figures are called? Indeed they are called congruent figures (â€˜congruentâ€™ means equal in all respects or figures whose shapes and sizes are both the same). Now, draw two circles of the same radius and place one on the other. What do you observe? They cover each other completely and we call them as congruent circles. Fig. 7.1 110 MATHEMA TICS File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 Repeat this activity by placing one square on the other with sides of the same measure (see Fig. 7.2) or by placing two equilateral triangles of equal sides on each other. You will observe that the squares are congruent to each other and so are the equilateral triangles. You may wonder why we are studying congruence. Y ou all must have seen the ice tray in your refrigerator. Observe that the moulds for making ice are all congruent. The cast used for moulding in the tray also has congruent depressions (may be all are rectangular or all circular or all triangular). So, whenever identical objects have to be produced, the concept of congruence is used in making the cast. Sometimes, you may find it difficult to replace the refill in your pen by a new one and this is so when the new refill is not of the same size as the one you want to remove. Obviously, if the two refills are identical or congruent, the new refill fits. So, you can find numerous examples where congruence of objects is applied in daily life situations. Can you think of some more examples of congruent figures? Now, which of the following figures are not congruent to the square in Fig 7.3 (i) : Fig. 7.3 The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i). Let us now discuss the congruence of two triangles. You already know that two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle. Fig. 7.2 Page 3 CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a triangle. (â€˜Triâ€™ means â€˜threeâ€™). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices. In Chapter 6, you have also studied some properties of triangles. In this chapter, you will study in details about the congruence of triangles, rules of congruence, some more properties of triangles and inequalities in a triangle. You have already verified most of these properties in earlier classes. We will now prove some of them. 7.2 Congruence of Triangles You must have observed that two copies of your photographs of the same size are identical. Similarly, two bangles of the same size, two A TM cards issued by the same bank are identical. You may recall that on placing a one rupee coin on another minted in the same year, they cover each other completely. Do you remember what such figures are called? Indeed they are called congruent figures (â€˜congruentâ€™ means equal in all respects or figures whose shapes and sizes are both the same). Now, draw two circles of the same radius and place one on the other. What do you observe? They cover each other completely and we call them as congruent circles. Fig. 7.1 110 MATHEMA TICS File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 Repeat this activity by placing one square on the other with sides of the same measure (see Fig. 7.2) or by placing two equilateral triangles of equal sides on each other. You will observe that the squares are congruent to each other and so are the equilateral triangles. You may wonder why we are studying congruence. Y ou all must have seen the ice tray in your refrigerator. Observe that the moulds for making ice are all congruent. The cast used for moulding in the tray also has congruent depressions (may be all are rectangular or all circular or all triangular). So, whenever identical objects have to be produced, the concept of congruence is used in making the cast. Sometimes, you may find it difficult to replace the refill in your pen by a new one and this is so when the new refill is not of the same size as the one you want to remove. Obviously, if the two refills are identical or congruent, the new refill fits. So, you can find numerous examples where congruence of objects is applied in daily life situations. Can you think of some more examples of congruent figures? Now, which of the following figures are not congruent to the square in Fig 7.3 (i) : Fig. 7.3 The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i). Let us now discuss the congruence of two triangles. You already know that two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle. Fig. 7.2 TRIANGLES 111 File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 Now, which of the triangles given below are congruent to triangle ABC in Fig. 7.4 (i)? Fig. 7.4 Cut out each of these triangles from Fig. 7.4 (ii) to (v) and turn them around and try to cover ? ABC. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent to ? ABC while ? TSU of Fig 7.4 (v) is not congruent to ? ABC. If ? PQR is congruent to ? ABC, we write ? PQR ? ? ABC. Notice that when ? PQR ? ? ABC, then sides of ? PQR fall on corresponding equal sides of ? ABC and so is the case for the angles. That is, PQ covers AB, QR covers BC and RP covers CA; ? P covers ? A, ? Q covers ? B and ? R covers ? C. Also, there is a one-one correspondence between the vertices. That is, P corresponds to A, Q to B, R to C and so on which is written as P ? A, Q ? B, R ? C Note that under this correspondence, ? PQR ? ? ABC; but it will not be correct to write ?QRP ? ? ABC. Similarly, for Fig. 7.4 (iii), Page 4 CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a triangle. (â€˜Triâ€™ means â€˜threeâ€™). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices. In Chapter 6, you have also studied some properties of triangles. In this chapter, you will study in details about the congruence of triangles, rules of congruence, some more properties of triangles and inequalities in a triangle. You have already verified most of these properties in earlier classes. We will now prove some of them. 7.2 Congruence of Triangles You must have observed that two copies of your photographs of the same size are identical. Similarly, two bangles of the same size, two A TM cards issued by the same bank are identical. You may recall that on placing a one rupee coin on another minted in the same year, they cover each other completely. Do you remember what such figures are called? Indeed they are called congruent figures (â€˜congruentâ€™ means equal in all respects or figures whose shapes and sizes are both the same). Now, draw two circles of the same radius and place one on the other. What do you observe? They cover each other completely and we call them as congruent circles. Fig. 7.1 110 MATHEMA TICS File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 Repeat this activity by placing one square on the other with sides of the same measure (see Fig. 7.2) or by placing two equilateral triangles of equal sides on each other. You will observe that the squares are congruent to each other and so are the equilateral triangles. You may wonder why we are studying congruence. Y ou all must have seen the ice tray in your refrigerator. Observe that the moulds for making ice are all congruent. The cast used for moulding in the tray also has congruent depressions (may be all are rectangular or all circular or all triangular). So, whenever identical objects have to be produced, the concept of congruence is used in making the cast. Sometimes, you may find it difficult to replace the refill in your pen by a new one and this is so when the new refill is not of the same size as the one you want to remove. Obviously, if the two refills are identical or congruent, the new refill fits. So, you can find numerous examples where congruence of objects is applied in daily life situations. Can you think of some more examples of congruent figures? Now, which of the following figures are not congruent to the square in Fig 7.3 (i) : Fig. 7.3 The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i). Let us now discuss the congruence of two triangles. You already know that two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle. Fig. 7.2 TRIANGLES 111 File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 Now, which of the triangles given below are congruent to triangle ABC in Fig. 7.4 (i)? Fig. 7.4 Cut out each of these triangles from Fig. 7.4 (ii) to (v) and turn them around and try to cover ? ABC. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent to ? ABC while ? TSU of Fig 7.4 (v) is not congruent to ? ABC. If ? PQR is congruent to ? ABC, we write ? PQR ? ? ABC. Notice that when ? PQR ? ? ABC, then sides of ? PQR fall on corresponding equal sides of ? ABC and so is the case for the angles. That is, PQ covers AB, QR covers BC and RP covers CA; ? P covers ? A, ? Q covers ? B and ? R covers ? C. Also, there is a one-one correspondence between the vertices. That is, P corresponds to A, Q to B, R to C and so on which is written as P ? A, Q ? B, R ? C Note that under this correspondence, ? PQR ? ? ABC; but it will not be correct to write ?QRP ? ? ABC. Similarly, for Fig. 7.4 (iii), 112 MATHEMA TICS File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 FD ? AB, DE ? BC and EF ? CA and F ? A, D ? B and E ? C So, ? FDE ? ? ABC but writing ? DEF ? ? ABC is not correct. Give the correspondence between the triangle in Fig. 7.4 (iv) and ? ABC. So, it is necessary to write the correspondence of vertices correctly for writing of congruence of triangles in symbolic form. Note that in congruent triangles corresponding parts are equal and we write in short â€˜CPCTâ€™ for corresponding parts of congruent triangles. 7.3 Criteria for Congruence of Triangles In earlier classes, you have learnt four criteria for congruence of triangles. Let us recall them. Draw two triangles with one side 3 cm. Are these triangles congruent? Observe that they are not congruent (see Fig. 7.5). Fig. 7.5 Now, draw two triangles with one side 4 cm and one angle 50° (see Fig. 7.6). Are they congruent? Fig. 7.6 Page 5 CHAPTER 7 TRIANGLES 7.1 Introduction You have studied about triangles and their various properties in your earlier classes. You know that a closed figure formed by three intersecting lines is called a triangle. (â€˜Triâ€™ means â€˜threeâ€™). A triangle has three sides, three angles and three vertices. For example, in triangle ABC, denoted as ? ABC (see Fig. 7.1); AB, BC, CA are the three sides, ? A, ? B, ? C are the three angles and A, B, C are three vertices. In Chapter 6, you have also studied some properties of triangles. In this chapter, you will study in details about the congruence of triangles, rules of congruence, some more properties of triangles and inequalities in a triangle. You have already verified most of these properties in earlier classes. We will now prove some of them. 7.2 Congruence of Triangles You must have observed that two copies of your photographs of the same size are identical. Similarly, two bangles of the same size, two A TM cards issued by the same bank are identical. You may recall that on placing a one rupee coin on another minted in the same year, they cover each other completely. Do you remember what such figures are called? Indeed they are called congruent figures (â€˜congruentâ€™ means equal in all respects or figures whose shapes and sizes are both the same). Now, draw two circles of the same radius and place one on the other. What do you observe? They cover each other completely and we call them as congruent circles. Fig. 7.1 110 MATHEMA TICS File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 Repeat this activity by placing one square on the other with sides of the same measure (see Fig. 7.2) or by placing two equilateral triangles of equal sides on each other. You will observe that the squares are congruent to each other and so are the equilateral triangles. You may wonder why we are studying congruence. Y ou all must have seen the ice tray in your refrigerator. Observe that the moulds for making ice are all congruent. The cast used for moulding in the tray also has congruent depressions (may be all are rectangular or all circular or all triangular). So, whenever identical objects have to be produced, the concept of congruence is used in making the cast. Sometimes, you may find it difficult to replace the refill in your pen by a new one and this is so when the new refill is not of the same size as the one you want to remove. Obviously, if the two refills are identical or congruent, the new refill fits. So, you can find numerous examples where congruence of objects is applied in daily life situations. Can you think of some more examples of congruent figures? Now, which of the following figures are not congruent to the square in Fig 7.3 (i) : Fig. 7.3 The large squares in Fig. 7.3 (ii) and (iii) are obviously not congruent to the one in Fig 7.3 (i), but the square in Fig 7.3 (iv) is congruent to the one given in Fig 7.3 (i). Let us now discuss the congruence of two triangles. You already know that two triangles are congruent if the sides and angles of one triangle are equal to the corresponding sides and angles of the other triangle. Fig. 7.2 TRIANGLES 111 File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 Now, which of the triangles given below are congruent to triangle ABC in Fig. 7.4 (i)? Fig. 7.4 Cut out each of these triangles from Fig. 7.4 (ii) to (v) and turn them around and try to cover ? ABC. Observe that triangles in Fig. 7.4 (ii), (iii) and (iv) are congruent to ? ABC while ? TSU of Fig 7.4 (v) is not congruent to ? ABC. If ? PQR is congruent to ? ABC, we write ? PQR ? ? ABC. Notice that when ? PQR ? ? ABC, then sides of ? PQR fall on corresponding equal sides of ? ABC and so is the case for the angles. That is, PQ covers AB, QR covers BC and RP covers CA; ? P covers ? A, ? Q covers ? B and ? R covers ? C. Also, there is a one-one correspondence between the vertices. That is, P corresponds to A, Q to B, R to C and so on which is written as P ? A, Q ? B, R ? C Note that under this correspondence, ? PQR ? ? ABC; but it will not be correct to write ?QRP ? ? ABC. Similarly, for Fig. 7.4 (iii), 112 MATHEMA TICS File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 FD ? AB, DE ? BC and EF ? CA and F ? A, D ? B and E ? C So, ? FDE ? ? ABC but writing ? DEF ? ? ABC is not correct. Give the correspondence between the triangle in Fig. 7.4 (iv) and ? ABC. So, it is necessary to write the correspondence of vertices correctly for writing of congruence of triangles in symbolic form. Note that in congruent triangles corresponding parts are equal and we write in short â€˜CPCTâ€™ for corresponding parts of congruent triangles. 7.3 Criteria for Congruence of Triangles In earlier classes, you have learnt four criteria for congruence of triangles. Let us recall them. Draw two triangles with one side 3 cm. Are these triangles congruent? Observe that they are not congruent (see Fig. 7.5). Fig. 7.5 Now, draw two triangles with one side 4 cm and one angle 50° (see Fig. 7.6). Are they congruent? Fig. 7.6 TRIANGLES 113 File Name : C:\Computer Station\Maths-IX\Chapter\Chap-7\Chap-7 (02-01-2006).PM65 See that these two triangles are not congruent. Repeat this activity with some more pairs of triangles. So, equality of one pair of sides or one pair of sides and one pair of angles is not sufficient to give us congruent triangles. What would happen if the other pair of arms (sides) of the equal angles are also equal? In Fig 7.7, BC = QR, ? B = ? Q and also, AB = PQ. Now, what can you say about congruence of ? ABC and ? PQR? Recall from your earlier classes that, in this case, the two triangles are congruent. Verify this for ? ABC and ? PQR in Fig. 7.7. Repeat this activity with other pairs of triangles. Do you observe that the equality of two sides and the included angle is enough for the congruence of triangles? Yes, it is enough. Fig. 7.7 This is the first criterion for congruence of triangles. Axiom 7.1 (SAS congruence rule) : Two triangles are congruent if two sides and the included angle of one triangle are equal to the sides and the included angle of the other triangle. This result cannot be proved with the help of previously known results and so it is accepted true as an axiom (see Appendix 1). Let us now take some examples. Example 1 : In Fig. 7.8, OA = OB and OD = OC. Show that (i) ? AOD ? ? BOC and (ii) AD || BC. Solution : (i) You may observe that in ? AOD and ? BOC, OA = OB (given) OD = OC Fig. 7.8 ? ? ?Read More