Page 1 ? Question:53 If one angle of a triangle is equal to the sum of the other two angles, then the triangle is a an isosceles triangle b an obtuse triangle c an equilateral triangle d a right triangle Solution: Let ?ABC be such that ?A = ?B + ?C. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? ?A + ?A = 180º ?A = ?B + ?C ? 2 ?A = 180º ? ?A = 90º Therefore, ?ABC is a right triangle. Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle. Hence, the correct answer is option d . Question:54 An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is a 70° b 55° c 35° d 27 1° 2 Solution: Let the measure of each of the two equal interior opposite angles of the triangle be x. In a triangle, the exterior angle is equal to the sum of the two interior opposite angles. ? x + x = 110° Page 2 ? Question:53 If one angle of a triangle is equal to the sum of the other two angles, then the triangle is a an isosceles triangle b an obtuse triangle c an equilateral triangle d a right triangle Solution: Let ?ABC be such that ?A = ?B + ?C. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? ?A + ?A = 180º ?A = ?B + ?C ? 2 ?A = 180º ? ?A = 90º Therefore, ?ABC is a right triangle. Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle. Hence, the correct answer is option d . Question:54 An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is a 70° b 55° c 35° d 27 1° 2 Solution: Let the measure of each of the two equal interior opposite angles of the triangle be x. In a triangle, the exterior angle is equal to the sum of the two interior opposite angles. ? x + x = 110° ? 2x = 110° ? x = 55° Thus, the measure of each of these equal angles is 55°. Hence, the correct answer is option b . Question:55 The angles of a triangle are in the ration 3 : 5 : 7. The triangle is a acute-angled b obtuse-angled c right-angled d an isosceles triangle Solution: a acute-angled Let the angles measure (3x)°, (5x)° and (7x)° . Then, 3x +5x +7x = 180° ? 15x = 180° ? x = 12° Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84° . Hence, the triangle is acute-angled. Question:56 If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be a 50° b 65° c 90° d 155° Solution: Let ?ABC be such that ?A = 130°. Here, BP is the bisector of ?B and CP is the bisector of ?C. ? ?ABP = ?PBC = 1 2 ?B .....1 Also, ?ACP = ?PCB = 1 2 ?C .....2 In ?ABC, ?A + ?B + ?C = 180° Anglesumproperty ? 130° + ?B + ?C = 180° ? ?B + ?C = 180° - 130° = 50° ? 1 2 ?B + 1 2 ?C = 1 2 × 50° = 25° ? ?PBC + ?PCB = 25° .....3 Using(1)and(2) In ?PBC, ?PBC + ?PCB + ?BPC = 180° Anglesumproperty ? 25° + ?BPC = 180° Page 3 ? Question:53 If one angle of a triangle is equal to the sum of the other two angles, then the triangle is a an isosceles triangle b an obtuse triangle c an equilateral triangle d a right triangle Solution: Let ?ABC be such that ?A = ?B + ?C. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? ?A + ?A = 180º ?A = ?B + ?C ? 2 ?A = 180º ? ?A = 90º Therefore, ?ABC is a right triangle. Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle. Hence, the correct answer is option d . Question:54 An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is a 70° b 55° c 35° d 27 1° 2 Solution: Let the measure of each of the two equal interior opposite angles of the triangle be x. In a triangle, the exterior angle is equal to the sum of the two interior opposite angles. ? x + x = 110° ? 2x = 110° ? x = 55° Thus, the measure of each of these equal angles is 55°. Hence, the correct answer is option b . Question:55 The angles of a triangle are in the ration 3 : 5 : 7. The triangle is a acute-angled b obtuse-angled c right-angled d an isosceles triangle Solution: a acute-angled Let the angles measure (3x)°, (5x)° and (7x)° . Then, 3x +5x +7x = 180° ? 15x = 180° ? x = 12° Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84° . Hence, the triangle is acute-angled. Question:56 If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be a 50° b 65° c 90° d 155° Solution: Let ?ABC be such that ?A = 130°. Here, BP is the bisector of ?B and CP is the bisector of ?C. ? ?ABP = ?PBC = 1 2 ?B .....1 Also, ?ACP = ?PCB = 1 2 ?C .....2 In ?ABC, ?A + ?B + ?C = 180° Anglesumproperty ? 130° + ?B + ?C = 180° ? ?B + ?C = 180° - 130° = 50° ? 1 2 ?B + 1 2 ?C = 1 2 × 50° = 25° ? ?PBC + ?PCB = 25° .....3 Using(1)and(2) In ?PBC, ?PBC + ?PCB + ?BPC = 180° Anglesumproperty ? 25° + ?BPC = 180° Using(3) ? ?BPC = 180° - 25° = 155° Thus, if one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles is 155°. Hence, the correct answer is option d . Question:57 In the given figure, AOB is a straight line. The value of x is a 12 b 15 c 20 d 25 Solution: It is given that, AOB is a straight line. ? 60º + (5xº + 3xº) = 180º Linearpair ? 8xº = 180º - 60º = 120º ? xº = 15º Thus, the value of x is 15. Hence, the correct answer is option b . Question:58 The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle of the triangle is a 12° b 100° c 80° d 60° Solution: Suppose ?ABC be such that ?A : ?B : ?C = 2 : 3 : 4. Let ?A = 2k, ?B = 3k and ?C = 4k, where k is some constant. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? 2k + 3k + 4k = 180º ? 9k = 180º ? k = 20º ? Measure of the largest angle = 4k = 4 × 20º = 80º Hence, the correct answer is option c . Question:59 In the given figure, ?OAB = 110° and ?BCD = 130° then ?ABC is equal to a 40° b 50° c 60° Page 4 ? Question:53 If one angle of a triangle is equal to the sum of the other two angles, then the triangle is a an isosceles triangle b an obtuse triangle c an equilateral triangle d a right triangle Solution: Let ?ABC be such that ?A = ?B + ?C. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? ?A + ?A = 180º ?A = ?B + ?C ? 2 ?A = 180º ? ?A = 90º Therefore, ?ABC is a right triangle. Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle. Hence, the correct answer is option d . Question:54 An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is a 70° b 55° c 35° d 27 1° 2 Solution: Let the measure of each of the two equal interior opposite angles of the triangle be x. In a triangle, the exterior angle is equal to the sum of the two interior opposite angles. ? x + x = 110° ? 2x = 110° ? x = 55° Thus, the measure of each of these equal angles is 55°. Hence, the correct answer is option b . Question:55 The angles of a triangle are in the ration 3 : 5 : 7. The triangle is a acute-angled b obtuse-angled c right-angled d an isosceles triangle Solution: a acute-angled Let the angles measure (3x)°, (5x)° and (7x)° . Then, 3x +5x +7x = 180° ? 15x = 180° ? x = 12° Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84° . Hence, the triangle is acute-angled. Question:56 If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be a 50° b 65° c 90° d 155° Solution: Let ?ABC be such that ?A = 130°. Here, BP is the bisector of ?B and CP is the bisector of ?C. ? ?ABP = ?PBC = 1 2 ?B .....1 Also, ?ACP = ?PCB = 1 2 ?C .....2 In ?ABC, ?A + ?B + ?C = 180° Anglesumproperty ? 130° + ?B + ?C = 180° ? ?B + ?C = 180° - 130° = 50° ? 1 2 ?B + 1 2 ?C = 1 2 × 50° = 25° ? ?PBC + ?PCB = 25° .....3 Using(1)and(2) In ?PBC, ?PBC + ?PCB + ?BPC = 180° Anglesumproperty ? 25° + ?BPC = 180° Using(3) ? ?BPC = 180° - 25° = 155° Thus, if one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles is 155°. Hence, the correct answer is option d . Question:57 In the given figure, AOB is a straight line. The value of x is a 12 b 15 c 20 d 25 Solution: It is given that, AOB is a straight line. ? 60º + (5xº + 3xº) = 180º Linearpair ? 8xº = 180º - 60º = 120º ? xº = 15º Thus, the value of x is 15. Hence, the correct answer is option b . Question:58 The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle of the triangle is a 12° b 100° c 80° d 60° Solution: Suppose ?ABC be such that ?A : ?B : ?C = 2 : 3 : 4. Let ?A = 2k, ?B = 3k and ?C = 4k, where k is some constant. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? 2k + 3k + 4k = 180º ? 9k = 180º ? k = 20º ? Measure of the largest angle = 4k = 4 × 20º = 80º Hence, the correct answer is option c . Question:59 In the given figure, ?OAB = 110° and ?BCD = 130° then ?ABC is equal to a 40° b 50° c 60° d 70° Solution: In the given figure, OA || CD. Construction: Extend OA such that it intersects BC at E. Now, OE || CD and BC is a transversal. ? ?AEC = ?BCD = 130° Pairofcorrespondingangles Also, ?OAB + ?BAE = 180° Linearpair ? 110° + ?BAE = 180° ? ?BAE = 180° - 110° = 70° In ?ABE, ?AEC = ?BAE + ?ABE Inatriangle, exteriorangleisequaltothesumoftwooppositeinteriorangles ? 130° = 70° + x° ? x° = 130° - 70° = 60° Thus, the measure of angle ?ABC is 60°. Hence, the correct answer is option c . Question:60 If two angles are complements of each other, then each angle is a an acute angle b an obtuse angle c a right angle d a reflex angle Solution: a an acute angle If two angles are complements of each other, that is, the sum of their measures is 90° , then each angle is an acute angle. Question:61 An angle which measures more than 180° but less than 360°, is called a an acute angle b an obtuse angle c a straight angle d a reflex angle Solution: An angle which measures more than 180° but less than 360° is called a reflex angle. Hence, the correct answer is option d . Question:62 The measure of an angle is five times its complement. The angle measures a 25° b 35° c 65° d 75° Solution: d 75° Page 5 ? Question:53 If one angle of a triangle is equal to the sum of the other two angles, then the triangle is a an isosceles triangle b an obtuse triangle c an equilateral triangle d a right triangle Solution: Let ?ABC be such that ?A = ?B + ?C. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? ?A + ?A = 180º ?A = ?B + ?C ? 2 ?A = 180º ? ?A = 90º Therefore, ?ABC is a right triangle. Thus, if one angle of a triangle is equal to the sum of the other two angles, then the triangle is a right triangle. Hence, the correct answer is option d . Question:54 An exterior angle of a triangle is 110° and its two interior opposite angles are equal. Each of these equal angles is a 70° b 55° c 35° d 27 1° 2 Solution: Let the measure of each of the two equal interior opposite angles of the triangle be x. In a triangle, the exterior angle is equal to the sum of the two interior opposite angles. ? x + x = 110° ? 2x = 110° ? x = 55° Thus, the measure of each of these equal angles is 55°. Hence, the correct answer is option b . Question:55 The angles of a triangle are in the ration 3 : 5 : 7. The triangle is a acute-angled b obtuse-angled c right-angled d an isosceles triangle Solution: a acute-angled Let the angles measure (3x)°, (5x)° and (7x)° . Then, 3x +5x +7x = 180° ? 15x = 180° ? x = 12° Therefore, the angles are 3(12)° = 36°, 5(12)° = 60° and 7(12)° = 84° . Hence, the triangle is acute-angled. Question:56 If one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles can be a 50° b 65° c 90° d 155° Solution: Let ?ABC be such that ?A = 130°. Here, BP is the bisector of ?B and CP is the bisector of ?C. ? ?ABP = ?PBC = 1 2 ?B .....1 Also, ?ACP = ?PCB = 1 2 ?C .....2 In ?ABC, ?A + ?B + ?C = 180° Anglesumproperty ? 130° + ?B + ?C = 180° ? ?B + ?C = 180° - 130° = 50° ? 1 2 ?B + 1 2 ?C = 1 2 × 50° = 25° ? ?PBC + ?PCB = 25° .....3 Using(1)and(2) In ?PBC, ?PBC + ?PCB + ?BPC = 180° Anglesumproperty ? 25° + ?BPC = 180° Using(3) ? ?BPC = 180° - 25° = 155° Thus, if one of the angles of a triangle is 130° then the angle between the bisectors of the other two angles is 155°. Hence, the correct answer is option d . Question:57 In the given figure, AOB is a straight line. The value of x is a 12 b 15 c 20 d 25 Solution: It is given that, AOB is a straight line. ? 60º + (5xº + 3xº) = 180º Linearpair ? 8xº = 180º - 60º = 120º ? xº = 15º Thus, the value of x is 15. Hence, the correct answer is option b . Question:58 The angles of a triangle are in the ratio 2 : 3 : 4. The largest angle of the triangle is a 12° b 100° c 80° d 60° Solution: Suppose ?ABC be such that ?A : ?B : ?C = 2 : 3 : 4. Let ?A = 2k, ?B = 3k and ?C = 4k, where k is some constant. In ?ABC, ?A + ?B + ?C = 180º Anglesumproperty ? 2k + 3k + 4k = 180º ? 9k = 180º ? k = 20º ? Measure of the largest angle = 4k = 4 × 20º = 80º Hence, the correct answer is option c . Question:59 In the given figure, ?OAB = 110° and ?BCD = 130° then ?ABC is equal to a 40° b 50° c 60° d 70° Solution: In the given figure, OA || CD. Construction: Extend OA such that it intersects BC at E. Now, OE || CD and BC is a transversal. ? ?AEC = ?BCD = 130° Pairofcorrespondingangles Also, ?OAB + ?BAE = 180° Linearpair ? 110° + ?BAE = 180° ? ?BAE = 180° - 110° = 70° In ?ABE, ?AEC = ?BAE + ?ABE Inatriangle, exteriorangleisequaltothesumoftwooppositeinteriorangles ? 130° = 70° + x° ? x° = 130° - 70° = 60° Thus, the measure of angle ?ABC is 60°. Hence, the correct answer is option c . Question:60 If two angles are complements of each other, then each angle is a an acute angle b an obtuse angle c a right angle d a reflex angle Solution: a an acute angle If two angles are complements of each other, that is, the sum of their measures is 90° , then each angle is an acute angle. Question:61 An angle which measures more than 180° but less than 360°, is called a an acute angle b an obtuse angle c a straight angle d a reflex angle Solution: An angle which measures more than 180° but less than 360° is called a reflex angle. Hence, the correct answer is option d . Question:62 The measure of an angle is five times its complement. The angle measures a 25° b 35° c 65° d 75° Solution: d 75° Let the measure of the required angle be x° . Then, the measure of its complement will be (90 -x)° . ? x = 5(90 -x) ? x = 450 -5x ? 6x = 450 ? x = 75 Question:63 Two complementary angles are such that twice the measure of one is equal to three times the measure of the other. The measure of larger angle is a 72° b 54° c 63° d 36° Solution: b 54° Let the measure of the required angle be x° . Then, the measure of its complement will be(90 -x)° . ? 2x = 3(90 -x) ? 2x = 270 -3x ? 5x = 270 ? x = 54 Question:64 In the given figure, AOB is a straight line. If ?AOC = 4x° and ?BOC = 5x°, then ?AOC = ? a 40° b 60° c 80° d 100° Solution: c 80° We have : ?AOC + ?BOC = 180° [Since AOB is a straight line] ? 4x +5x = 180° ? 9x = 180° ? x = 20° ? ?AOC = 4 ×20° = 80° Question:65 In the given figure, AOB is a straight line. If ?AOC = (3x + 10)° and ?BOC (4x - 26)°, then ?BOC = ? a 96° b 86° c 76° d 106° Solution: b 86° We have : ?AOC + ?BOC = 180° [Since AOB is a straight line ] ? 3x +10 +4x -26 = 180° ? 7x = 196° ? x = 28° ? ?BOC = [4 ×28 -26] ° Hence, ?BOC = 86°. Question:66 In the given figure, AOB is a straight line. If ?AOC = (3x - 10)°, ?COD = 50° and ?BOD = (x + 20)°, then ?AOC = ? a 40° b 60° c 80° d 50° Solution: c 80° We have : ?AOC + ?COD + ?BOD = 180° [Since AOB is a straight line ] ? 3x -10 +50 +x +20 = 180 ? 4x = 120 ? x = 30Read More

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