NCERT Solutions for class 7 Maths Chapter 7: Congruence of TrianglesExercise 7.11. Complete the following statements: (a) Two line segments are congruent if they have the same length. Answer: they have the same length Explanation: If the length of the two lines is equal, they are termed as congruent. Similarly, if two line segments are congruent, they have the equal length. (b) Among two congruent angles, one has a measure of 70°; the measure of the other angle is 70°. Answer: 70° Explanation: If the measure of two angles is equal, they are termed as congruent. Similarly, if two angles are congruent, they have the equal measure. Here, if one angle has a measure of 70°, the other congruent angle is also equal to 70°. &(c) When we write ∠A = ∠B, we actually mean m∠A = m∠ B. Answer: m∠A = m∠ B Explanation: Here, m means measure. ∠A = ∠B means that the value or the measure of the two angles are equal. If the measure of ∠A is 50°, the measure of ∠B is also equal to 50°. 2. Give any two real-life examples for congruent shapes. Answer: The real examples for the congruent shapes are as follows:
3. If ∆ABC ≅ ∆FED under the correspondence ABC ↔ FED, write all the corresponding congruent parts of the triangles. Answer: ∠A ↔ ∠F, ∠B ↔ ∠E, ∠C ↔ ∠D; AB ↔FE, BC ↔ED, AC↔ FD Explanation: Two triangles are said to be congruent if its three sides and all the three angles are congruent. Let's consider these two triangles. Since these triangles are congruent, The angles of these two triangles are congruent.
The sides of these two triangles are also congruent.
4. If ∆DEF ≅ ∆BCA, write the part(s) of ∆BCA that correspond to (i) ∠E Answer: ∠C Explanation: Two triangles are said to be congruent if its three sides and all the three angles are congruent. ∠E of the triangle DEF corresponds to the ∠C of the triangle BCA. (ii) EF Answer: CA Explanation: Two triangles are said to be congruent if its three sides and all the three angles are congruent. EF of the triangle DEF corresponds to the CA of the triangle BCA. (iii) ∠F Answer: ∠A Explanation: Two triangles are said to be congruent if its three sides and all the three angles are congruent. ∠F of the triangle DEF corresponds to the ∠A of the triangle BCA. (iv) DF Answer: BA Explanation: Two triangles are said to be congruent if its three sides and all the three angles are congruent. DF of the triangle DEF corresponds to the BA of the triangle BCA. Exercise 7.21. Which congruence criterion do you use in the following? (a) Given: AC = DF, AB = DE, BC = EF. So, ∆ABC ≅ ∆DEF Answer: SSS Congruence Criterion Explanation: All the three given parts are the sides of the two triangles. Hence, these triangles are congruent according to the SSS criteria. (b) Given: ZX = RP, RQ = ZY, ∠PRQ = ∠XZY. So, ∆PQR ≅ ∆XYZ Answer: SAS Congruence Criterion Explanation: The three given parts are the two sides and one angle of the two triangles. Hence, these triangles are congruent according to the SAS criteria. (c) Given: ∠MLN = ∠FGH, ∠NML = ∠GFH, ML = FG. So, ∆LMN ≅ ∆GFH Answer: ASA Congruence Criterion Explanation: The three given parts are the two angles and one side of the two triangles. Hence, these triangles are congruent according to the ASA criteria. (d) Given: EB = DB, AE = BC, ∠A = ∠C = 90°. So, ∆ABE ≅ ∆CDB Answer: RHS Congruence Criterion Explanation: When the two triangles are congruent with two equal sides and one right angle, the congruency criteria is termed as RHS Congruence Criterion. Hence, these triangles are congruent according to the RHS criteria. 2. You want to show that ∆ART ≅ ∆PEN, (a) If you have to use SSS criterion, then you need to show (i) AR = Answer: AR = PE Explanation: According to the SSS Congruency, the side which seems its exact copy is the equivalent side. Here, PE is the equivalent side of the AR. (ii) RT = Answer: RT = EN Explanation: According to the SSS Congruency, the side which seems its exact copy is the equivalent side. Here, EN is the equivalent side of the RT. (iii) AT = Answer: AT = PN Explanation: According to the SSS Congruency, the side which seems its exact copy is the equivalent side. Here, PN is the equivalent side of the AT. (b) If it is given that ∠T = ∠N and you are to use SAS criterion, you need to have (i) RT = Answer: RT = EN Explanation: According to the SAS Congruency, the side which seems its exact copy is the equivalent side. Here, EN is the equivalent side of the RT. and (ii) PN = Answer: PN = AT Explanation: According to the SAS Congruency, the side which seems its exact copy is the equivalent side. Here, AT is the equivalent side of the PN. (c) If it is given that AT = PN and you are to use ASA criterion, you need to have (i) ? Answer: ∠RAT = ∠EPN Explanation: According to the ASA Congruency, one side and two angles need to be congruent. One side is already given. The other two equal parts will be its angles. The equivalent angles are ∠RAT = ∠EPN. (ii) ? Answer: ∠ATR = ∠PNE Explanation: According to the ASA Congruency, one side and two angles need to be congruent. One side is already given. The other two equal parts will be its angles. One is already defined above. The other equivalent angles are ∠ATR = ∠PNE. Thus, the three equal parts according to the ASA criteria are: AT = PN (Given) ∠RAT = ∠EPN ∠RAT = ∠EPN 3. You have to show that ∆AMP ≅ ∆AMQ. In the following proof, supply the missing reasons.
4. In ∆ABC, ∠A = 30°, ∠B = 40° and ∠C = 110°. In ∆PQR, ∠P = 30° , ∠Q = 40° and ∠R = 110° A student says that ∆ABC ≅ ∆PQR by AAA congruence criterion. Is he justified? Why or why not? Answer: No Explanation: There are no such criteria of AAA congruency. Two triangles with equal three angles need not to be congruent. In such a case, one triangle can be the enlarged copy of the other triangle. They cannot be as congruent because of their different dimensions. They can only be termed as congruent if they are the exact copies of each other. Hence, the given statement of the student is not justified. 5. In the figure, the two triangles are congruent. The corresponding parts are marked. We can write ∆RAT ≅ ? Answer: ∆RAT ≅ ∆WON Explanation: In the given two triangles, the pairs of equal parts are: AT = ON (Given) AR = OW (Given) ∠T = ∠N (Given) ∠R = ∠W (Given) ∠A = ∠O (Given) Hence, the two triangles are congruent. 6. Complete the congruence statement: ∆BCA ≅ ? Answer: ∆BCA ≅ ∆BTA In the given two triangles, the three pairs of equal parts are: BT = BC (Given) AT = AC (Given) ∠TAB = ∠CAB (right angle) Hence, the two triangles are congruent according to the SAS criteria. ∆QRS ≅ ? Answer: ∆ QRS ≅ ∆TPQ In the given two triangles, the three pairs of equal parts are: RS = PQ (Given) QS = TQ (Given) ∠QSR = ∠PQT Hence, the two triangles are congruent according to the SAS criteria. 7. In a squared sheet, draw two triangles of equal areas such that (i) The triangles are congruent. Answer: Let's draw the two congruent triangles on the square sheet. In the given two triangles, the three pairs of equal parts are: BC = EF (Side) AB = DE (Side) AC = DF (Side) Hence, the two triangles are congruent according to the SSS criteria. (ii) The triangles are not congruent. What can you say about their perimeters? Answer: Let's draw the two non-congruent triangles on the square sheet. Perimeter is equal to the sum of the three sides of the triangle. Since the two triangles are not congruent, their perimeter will be unequal. AB + BC + CA is not equal to DE + EF + DF 8. Draw a rough sketch of two triangles such that they have five pairs of congruent parts but still the triangles is not congruent. Answer: In the given two triangles, the five pairs of equal parts are: BC = EF (Side) AB = DE (Side) ∠C = ∠F ∠B = ∠E ∠A = ∠D But, it does not follow any criterion for the congruency. Hence, the given two triangles are not congruent. 9. If ∆ABC and ∆PQR are to be congruent, name one additional pair of corresponding parts. What criterion did you use? Answer: BC = QR, ASA Congruence criterion Explanation: If ∆ABC and ∆PQR are to be congruent, side DC of the triangle ABC and side QR of the triangle PQR will be equal. In the given two triangles, the three pairs of equal parts are: BC = QR ∠C = ∠R (Given) ∠B = ∠Q (Right angle) Hence, the two triangles are congruent as per the ASA criteria. 10. Explain, why ∆ABC ≅ ∆FED. Answer: ASA Congruence criterion Explanation: In the given two triangles, the three pairs of equal parts are: BC = DE (Given) ∠A = ∠F (Given) ∠B = ∠E (Right angle) Hence, the two triangles are congruent as per the ASA criteria. |