Questions and Answers #3 Congruence

Questions and Answers Sheet 3 Congruence Question #1 In the roof truss shown, AB=8 and 𝑚∠𝐻𝐴𝐹 = 37∘ . Find: AH Given: AB=8 and 𝑚∠𝐻𝐴𝐹 = 37∘ . The roof truss has line symmetry. Figure(1) Answer: Approach: (1) If a figure has line symmetry then one side of the symmetric line is congruent to the other side of the line. (2) If two triangles are congruent to each other then all sides of the triangle are congruent. Calculation: ⃗⃗⃗⃗⃗ . The roof truss has line symmetry about line 𝐴𝐸 So, △ 𝐴𝐵𝐷 ≅△ 𝐴𝐻𝐹 . Use transitive property of congruence. △ 𝐴𝐵𝐷 ≅△ 𝐴𝐻𝐹 and AH, and AB are the sides of congruent triangles. So, AH=AB Substitute 8 for AH in order to find the value of AH. AH=8 Conclusion: Therefore, the value of AH is 8. Question #2 In the roof truss shown, AB=8 and 𝑚∠𝐻𝐴𝐹 = 37∘ . Find: 𝑚∠𝐵𝐴𝐷 Given: AB=8 and 𝑚∠𝐻𝐴𝐹 = 37∘ . The roof truss has line symmetry. Figure(1) Answer: Concept: If two triangles are congruent to each other then all angles of the triangle are congruent. Calculation: Use transitive property of congruence. △ 𝐴𝐵𝐷 ≅△ AHF and 𝑚∠𝐻𝐴𝐹, and 𝑚∠𝐵𝐴𝐷 are the angles of congruent triangles. So, 𝑚∠𝐵𝐴𝐷 = 𝑚∠𝐻𝐴𝐹 Substitute 37∘ for m\angle HAF in order to find the value of 𝑚∠𝐵𝐴𝐷 . 𝑚∠𝐵𝐴𝐷 = 37∘ Conclusion: Therefore, the value of 𝑚∠𝐵𝐴𝐷 is 37∘ . Question #3 In the roof truss shown, AB=8 and 𝑚∠𝐻𝐴𝐹 = 37∘ . Find: 𝑚∠𝐴𝐷𝐵 Given: AB=8 and 𝑚∠𝐻𝐴𝐹 = 37∘ . The roof truss has line symmetry. Figure(1) Answer: Approach: The sum of measure of three angles in a triangle is 180∘ . Calculation: Equate the sum of measure of three angles of the triangle to 180∘ . 𝑚∠𝐵𝐴𝐷 + 𝑚∠𝐴𝐵𝐷 + 𝑚∠𝐴𝐷𝐵 = 180∘ . . (1) Substitute 37∘ for 𝑚∠𝐵𝐴𝐷 and 90∘ for 𝑚∠𝐴𝐵𝐷 in equation (1) in order to find the value of 𝑚∠𝐴𝐷𝐵 . 37∘ + 90∘ + 𝑚∠𝐴𝐷𝐵 = 180∘ 𝑚∠𝐴𝐷𝐵 = 180∘ — 37∘ — 90∘ = 53∘ Therefore, the value of 𝑚∠𝐴𝐷𝐵 is 53∘ . Conclusion: The value of 𝑚∠𝐴𝐷𝐵 is 53∘ . Question #4 The value of the operation [9][6] in 𝑍10 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If xand yare integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [x] [y] =[xy]. Then, the value of the expression becomes as follows. [9][6] = [9 × 6] = [54] Since 54  4 ( mod1) 0 . Thus [9][6]=[4] in Therefore, the value of the expression is [4]. 𝑍10 . Question #5 The value of the operation [8][7] in 𝑍6 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [x] + [y] = [x+y]. Then, the value of the expression becomes as follows. [8][7] = [8 × 7] = [56] Since 56  2 ( mod 6) . Thus [8][7]=[2] in Therefore, the value of the expression is [2]. Question #6 To prove: △ 𝑌𝑋𝑍 ≅△ 𝑊𝑍𝑋 Given information: ¯{𝑌𝑋} ≅ ¯{𝑊𝑍} & ¯{𝑌𝑋}||𝑍𝑊 𝑍6 . Answer: Proof: Since ¯{𝑌𝑋} ≅ ¯{𝑊𝑍} & ¯{𝑌𝑋}||¯{𝑍𝑊} By reflexive property of congruence, 𝑌𝑍 ≅ 𝑌𝑍 As alternate interior angles of two parallel lines are congruent, ∠𝑊𝑍𝑋 ≅ ∠𝑌𝑋𝑍 By properties of Side Angle Side congruence postulate, Therefore, △ 𝑌𝑋𝑍 ≅△ 𝑊𝑍𝑋 Question #7 To decide: Whether the given information is enough to prove that △ 𝑊𝑋𝑍 ≅△ 𝑌𝑍𝑋 using SAS Congruency Theorem. If so write a proof if not then explain the reason. Given: The given figure is as follows. Answer: Calculation: Recollect the HL Congruence Theorem: If the hypotenuse and a leg of a right triangle are congruent to the hypotenuse and a leg of second right triangle, then the two triangles are congruent. From the given figure, the triangles △ 𝑊𝑋𝑍 and △ 𝑌𝑍𝑋 are two right triangles with 𝑊𝑋 ≅ 𝑌𝑍. By Reflexive Property of Congruence theorem, 𝑋𝑍 ≅ 𝑍𝑋 . Therefore, the △ 𝑊𝑋𝑍 and △ 𝑌𝑍𝑋 are congruent. Question #8 The value of the operation [12]+[25] in 𝑍7 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [x] + [y] = [x+y]. Then, the value of the expression becomes as follows. [12] + [25] = [12 + 25] = [37] Since ( mod 7 ) . Thus [12] + [25] = [2] in 37  2 𝑍7 . Therefore, the value of the expression is [2]. Question #9 The value of the expression [5]+[10] in 𝑍11 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [x] + [y] = [x+y]. Then, the value of the expression becomes as follows. [5] + [10] = [5 + 10] = [15] Since ( mod1)1 . Thus [5] + [10] = [4] in 15  4 𝑍11 . Therefore, the value of the expression is [4]. Question #10 The value of the operation [5][12] in 𝑍8 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [x] [y] = [xy]. Then, the value of the expression becomes as follows. [5][12] = [5 × 12] = [60] Since 60  4 ( mod8) . Thus [5] [12] = [4] in 𝑍8 . Therefore, the value of the expression is [5]. Question #11 Whether pq ( mod m ) , when p=29, q=-34, and m=7. Answer: Definition used: Congruence Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as ( mod m ) . This relation is called as congruence modulo x= y m. Calculation: Given numbers are p = 29, q = 34, and m =7. For being congruence 29 - (-34) = 63 has to divisible by 7. That is, 63 = 9*7+0. Here, the remainder is 0. Therefore, 63 is divisible by 7. Therefore, When p = 29,q = -34, and m = 7, pq ( mod m ) . Question #12 The value of the operation [16]+[9] in 𝑍6 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as ( mod m ) . This relation is called as congruence modulo x= y m. Calculation: It is known that, [x] + [y] = [x+y]. Then, the value of the expression becomes as follows. [16] + [9] = [16 + 9] = [25] Since 25  1 ( mod 6) . Thus [16] + [9] = [1] in 𝑍6 . Therefore, the value of the expression is [1]. Question #13 The value of the operation [8][11] in 𝑍5 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [x] [y] = [xy]. Then, the value of the expression becomes as follows. [8][11] = [8 × 11] = [88] Since 88 = 3 ( mod 5) . Thus [8][11] = [3] in Therefore, the value of the expression is [3]. Question #14 𝑍5 . Solve the following congruence. x2  7 ( mod 2 ) 9 Answer: Step 1 The value for x = 6. Find the remainder r using the congruence of modulo. x 2 mod 29  r w 7 𝑚𝑜𝑑 29 ≡ 𝑟 So, Divisor=29 Dividend=7 Remaider=7 Thus, r=7 Sep 2 Now, calculate the value for x using the quotient as q = 1. 𝑥 2 𝑚𝑜𝑑 29 ≡ 𝑟 𝑥 2 𝑚𝑜𝑑 29 ≡ 7 29 × 𝑞 + 𝑟 = 𝑥 2 For q= 1, (29 × 1) + 7 = 𝑥 2 29 + 7 = 𝑥 2 𝑥 2 = 36 𝑥=6 Hence, x=6 Question #15 The value of the operation [3][20] in 𝑍11 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [𝑥][𝑦] = [𝑥𝑦]. Then, the value of the expression becomes as follows. [3][20] = [3 × 20] = [60] Since 60 ≡ 5(𝑚𝑜𝑑 11). Thus[3][20] = [5]in 𝑍11 . Therefore, the value of the expression is [5]. Question #16 The value of the operation [9] + [8] in 𝑍13 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as 𝑥 = 𝑦(𝑚𝑜𝑑𝑚). This relation is called as congruence modulo m. Calculation: It is known that, [𝑥] + [𝑦] = [𝑥 + 𝑦]. Then, the value of the expression becomes as follows. [9] + [8] = [9 + 8]=[17] Since 17 ≡ 4 (mod 13). Thus [9] + [8] = [4] in 𝑍13 . Therefore, the value of the expression is [4]. Question #17 To prove: ¯{𝐶𝐷}||𝐵𝐸 Given: The diagram: The line 𝐵𝐸 bisects ∠𝐷𝐵𝐴 ∠3 ≅ ∠1 Answer: Concept Used: Transitive property of congruence: if a\cong b and 𝑏 ≅ 𝑐 , then 𝑎 ≅ 𝑐. Theorem 3-5: If two lines are cut by a transversal and alternate interior angles are congruent, then the lines are parallel. It is given that, the line 𝐵𝐸 bisects ∠𝐷𝐵𝐴 , this implies that, 𝑚∠𝐷𝐵𝐸 = 𝑚∠𝐸𝐵𝐴 𝑚∠2 = 𝑚∠3 That is, ∠2 ≅ ∠3 Now applying the transitive property of congruence on ∠1 ≅ ∠3 and ∠3 ≅ ∠2 , it can be said that ∠1 ≅ ∠2 From the above figure, it can be observed that the angles ∠1 𝑎𝑛𝑑 ∠2 are the alternate interior angles which are formed when the lines 𝐶𝐷 and 𝐵𝐸 are cut by the transversal 𝐷𝐵 . And since ∠1 ≅ ∠2 , so by the theorem 3-5, it can be said that the lines 𝐶𝐷 and 𝐵𝐸 are parallel. Question #18 The value of the operation [8]+[6] in 𝑍12 and to write the answer in the form [r] with 0 ≤ 𝑟 < 𝑚. Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if x - y is divisible by m. It can be represented as x= y ( mod m ) . This relation is called as congruence modulo m. Calculation: It is known that, [𝑥] + [𝑦] = [𝑥 + 𝑦]. Then, the value of the expression becomes as follows. [8] + [6] = [8 + 6] = [14] Since 14 ≡ 2 (𝑚𝑜𝑑12). Thus [8] + [6] = [2] in 𝑍12 . Therefore, the value of the expression is [2]. Question #19 The value of the expression [9] + [11] in 𝑍15 and to write the answer in the form [𝑟] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if 𝑥 − 𝑦 is divisible by m. It can be represented as 𝑥 = 𝑦(𝑚𝑜𝑑 𝑚). This relation is called as congruence modulo m. Calculation: It is known that, [𝑥] + [𝑦] = [𝑥 + 𝑦]. Then, the value of the expression becomes as follows. [9] + [11] = [9 + 11] = [20] Since 20  5 ( mod1) 5 . Thus [9] + [11] = [5] in 𝑍15 . Therefore, the value of the expression is [5]. Question #20 The value of the operation [43] + [31] in 𝑍22 and to write the answer in the form [𝑟] with 0 ≤ 𝑟 < 𝑚 . Answer: Definition used: Congruence. Let m be an integer greater than 1. If x and y are integers, then x is congruent to y modulo m if 𝑥 — 𝑦 is divisible by m. It can be represented as m. Calculation: x= y ( mod m ) . This relation is called as congruence modulo It is known that, [𝑥] + [𝑦] = [𝑥 + 𝑦]. Then, the value of the expression becomes as follows. [43] + [31] = [43 + 31] = [74] Since 74 = 8 ( mod 2 ) 2 . Thus [43] + [31] = [8] in Therefore, the value of the expression is [8]. 𝑍22 .