Class 9th Mathematics Term 1 Tamilnadu Board Solution
Exercise 4.1- Find the complement of each of the following angles. i. 63° ii. 24° iii. 48° iv.…
- Find the supplement of each of the following angles. i. 58° ii. 148° iii. 120°…
- Find the value of x in the following figures. i. ii.
- The angle which is two times its complement. Find the angles in each of the…
- The angle which is four times its supplement. Find the angles in each of the…
- The angles whose supplement is four times its complement. Find the angles in…
- The angle whose complement is one sixth of its supplement. Find the angles in…
- Supplementary angles are in the ratio 4:5. Find the angles in each of the…
- Two complementary angles are in the ratio 3:2. Find the angles in each of the…
- Find the values of x, y in the following figures.
- Find the values of x, y in the following figures.
- Find the values of x, y in the following figures.
- Let l1 || l2 and m1 is a transversal. If ∠F = 65°, find the measure of each of…
- For what value of x will l1 and l2 be parallel lines. i. ii.
- The angles of a triangle are in the ratio of 1:2:3. Find the measure of each…
- In ΔABC, ∠A + ∠B = 70° and ∠B + ∠C = 135°. Find the measure of each angle of the…
- In the given figure at right, side BC of ΔABC is produced to D. Find ∠A and ∠C.…
Exercise 4.3- If an angle is equal to one third of its supplement, its measure is equal toA.…
- In the given figure, OP bisect ∠ BOC and OQ bisect ∠ AOC. Then ∠ POQ is equal to…
- The complement of an angle exceeds the angle by 60°. Then the angle is equal…
- Find the measure of an angle, if six times of its complement is 12° less than…
- In the given figure, ∠ B:∠ C = 2:3, find ∠ B A. 120° B. 52° C. 78° D. 130°…
- ABCD is a parallelogram, E is the mid - point of AB and CE bisects ∠ BCD. Then ∠…
- Find the complement of each of the following angles. i. 63° ii. 24° iii. 48° iv.…
- Find the supplement of each of the following angles. i. 58° ii. 148° iii. 120°…
- Find the value of x in the following figures. i. ii.
- The angle which is two times its complement. Find the angles in each of the…
- The angle which is four times its supplement. Find the angles in each of the…
- The angles whose supplement is four times its complement. Find the angles in…
- The angle whose complement is one sixth of its supplement. Find the angles in…
- Supplementary angles are in the ratio 4:5. Find the angles in each of the…
- Two complementary angles are in the ratio 3:2. Find the angles in each of the…
- Find the values of x, y in the following figures.
- Find the values of x, y in the following figures.
- Find the values of x, y in the following figures.
- Let l1 || l2 and m1 is a transversal. If ∠F = 65°, find the measure of each of…
- For what value of x will l1 and l2 be parallel lines. i. ii.
- The angles of a triangle are in the ratio of 1:2:3. Find the measure of each…
- In ΔABC, ∠A + ∠B = 70° and ∠B + ∠C = 135°. Find the measure of each angle of the…
- In the given figure at right, side BC of ΔABC is produced to D. Find ∠A and ∠C.…
- If an angle is equal to one third of its supplement, its measure is equal toA.…
- In the given figure, OP bisect ∠ BOC and OQ bisect ∠ AOC. Then ∠ POQ is equal to…
- The complement of an angle exceeds the angle by 60°. Then the angle is equal…
- Find the measure of an angle, if six times of its complement is 12° less than…
- In the given figure, ∠ B:∠ C = 2:3, find ∠ B A. 120° B. 52° C. 78° D. 130°…
- ABCD is a parallelogram, E is the mid - point of AB and CE bisects ∠ BCD. Then ∠…
Exercise 4.1
Question 1.Find the complement of each of the following angles.
i. 63° ii. 24°
iii. 48° iv. 35°
v. 20°
Answer:i. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 63° = 90° - 63° = 27°
ii. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 24° = 90° - 24° = 66°
iii. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 48° = 90° - 48° = 42°
iv. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 35° = 90° - 35° = 55°
v. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 20° = 90° - 20° = 70°
Question 2.Find the supplement of each of the following angles.
i. 58° ii. 148°
iii. 120° iv. 40°
v. 100°
Answer:i. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 58° = 180° - 58° = 122°
ii. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 148° = 180° - 148° = 32°
iii. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 120° = 180° - 120° = 60°
iv. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 40° = 180° - 40° = 140°
v. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 100° = 180° - 100° = 80°
Question 3.Find the value of x in the following figures.
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,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)
Answer:i. In the given figure,
AB is a straight line, hence all the angles formed on it are supplementary.
⇒ (x – 20)° + x° + 40° = 180°
⇒ 2x + 20° = 180°
⇒ 2x = 180° - 20°
⇒ 2x = 160°
⇒ x = 80°
ii. In the given figure,
AB is a straight line, hence all the angles formed on it are supplementary.
⇒ (x + 30)° + (115 – x)° + x° = 180°
⇒ x + 145° = 180°
⇒ x = 180° - 145°
⇒ x = 35°
Question 4.Find the angles in each of the following.
The angle which is two times its complement.
Answer:Let the complement of an angle be x°
According to the question,
Angle = 2x°
We know that two angles are said to be complementary if their sum is 90°.
So, 2x° + x° = 90°
⇒ 3x° = 90°
⇒ x° = 30°
∴ Angle = 2x° = 2× 30° = 60°, Compliment = x° = 30°
Question 5.Find the angles in each of the following.
The angle which is four times its supplement.
Answer:Let the supplement of an angle be x°
According to the question,
Angle = 4x°
We know that two angles are said to be supplementary if their sum is 180°.
So, 4x° + x° = 180°
⇒ 5x° = 180°
⇒ x° = 36°
∴ Angle = 4x° = 4× 36° = 144°, Compliment = x° = 36°
Question 6.Find the angles in each of the following.
The angles whose supplement is four times its complement.
Answer:Let the compliment of an angle be x°
According to the question,
Supplement of the angle = 4x°
We know that two angles are said to be supple mentary if their sum is 180°.
So, required angle = 180° - 4x° ..(I)
Also, if the two angles are complimentary their sum is 90°.
⇒ required angle = 90° - x° …(II)
Equating I and II,
90° - x° = 180° - 4x°
⇒ 3x° = 90°
⇒ x° = 30°
Hence, required angle = 90° - x° = 90° - 30° = 60°
Question 7.Find the angles in each of the following.
The angle whose complement is one sixth of its supplement.
Answer:Let the supplement of an angle be x°
According to the question,
Compliment of the angle ![](data:image/png;base64,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)
We know that two angles are said to be supple-mentary if their sum is 180°.
So,
..(I)
Also, if the two angles are complimentary their sum is 90°.
…(II)
Equating I and II,
![](data:image/png;base64,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)
![](data:image/png;base64,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)
![](data:image/png;base64,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)
⇒ x° = 108°
Hence, required angle = 180° - x° = 180° - 108° = 72°
Question 8.Find the angles in each of the following.
Supplementary angles are in the ratio 4:5.
Answer:Let the supplementary angles be 4x and 5x
We know that two angles are said to be supple mentary if their sum is 180°.
⇒ 4x + 5x = 180°
⇒ 9x = 180°
⇒ x = 20°
So, the supplementary angles will be 4× 20° = 80° and 5× 20° = 100°
Question 9.Find the angles in each of the following.
Two complementary angles are in the ratio 3:2.
Answer:Let the complimentary angles be 3x and 2x
We know that two angles are said to be complimentary if their sum is 90°.
⇒ 2x + 3x = 90°
⇒ 5x = 90°
⇒ x = 18°
So, the complimentary angles will be 2×18° = 36° and 3×18° = 54°
Question 10.Find the values of x, y in the following figures.
![](data:image/png;base64,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)
Answer:Given: ∠ADC = 3x and ∠BDC = 2x
Here AB is the straight line.
∴ ∠ADC and ∠BDC are linear pair.
⇒ ∠ADC + ∠BDC = 180°
⇒ 3x + 2x = 180°
⇒ 5x = 180°
⇒ x = 36°
Question 11.Find the values of x, y in the following figures.
![](data:image/png;base64,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)
Answer:Given: ∠ADC = (3x + 5)° and ∠BDC = (2x – 25)°
Here AB is the straight line.
∴ ∠ADC and ∠BDC are linear pair.
⇒ ∠ADC + ∠BDC = 180°
⇒ 3x + 5 + 2x - 25 = 180°
⇒ 5x - 20 = 180°
⇒ 5x = 200°
⇒ x = 40°
Question 12.Find the values of x, y in the following figures.
![](data:image/png;base64,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)
Answer:Given: ∠AOD = y° ,∠DOC = 90°,∠COB = x° ,∠AOE = 3x and ∠BOE = 60°
Here AB is the straight line.
∴ ∠AOE and ∠BDC are linear pair.
⇒ ∠ADC + ∠BOE = 180°
⇒ 3x + 60° = 180°
⇒ 3x = 120°
⇒ x = 40°
Also, ∠AOD, ∠DOC and ∠BOC are linear pair.
⇒ ∠AOD + ∠DOC + ∠BOC = 180°
⇒ y + 90° + x° = 180°
⇒ y = 90° - 40°
⇒ y = 50°
Question 13.Let l1 || l2 and m1 is a transversal. If ∠F = 65°, find the measure of each of the remaining angles.
![](data:image/png;base64,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)
Answer:Given: ∠F = 65°
∠B = ∠F = 65° {Corresponding angle}
∠H = ∠F = 65° {vertically opposite angle}
∠D = ∠B = 65° {vertically opposite angle}
Also, ∠C + ∠F = 180° {co-interior angles are supplementary}
⇒ ∠C = 180° - 65°
⇒ ∠C = 115°
∠E = ∠C = 115° {Alternate interior angle}
∠G = ∠C = 115° {Corresponding angle}
∠A = ∠G = 115° {Alternate exterior angle}
Question 14.For what value of x will l1 and l2 be parallel lines.
i. ![](data:image/png;base64,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)
ii. ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAKAAAABvCAIAAADkE1wbAAAAAXNSR0IArs4c6QAAAAlwSFlzAAAOxAAADsMB2mqY3AAADGZJREFUeF7tnWtsXMUVx+M4iWkS/KCFD1XtGAUJCZLGpf1CTImjfsiDkvgTSiCJg4Rag/tIP7QRtBIgVRRatbglAdNWIpAQ06pSHdo4iVTJbmuHqlLAYFOpgijGblVEmti7thPb8aM/+7g3633cnd29d+7c3VmtVtfXc8+cOf85Z2bOnDm3aHZ2don95K8EluZv02zL5iRgAc7zfmABtgDnuQTyvHlWgy3AeS6BwJr38kstfb29fldvNdhvCSen/8bx1l+2tKxbv97v6ovsOthvESfS/2ZT06mT7dyvrKrq+MuffWXAAuyreFMS/+KGmq81Nn790Ua/q7cm2l8JT/VcnGz7cHT/6as/+rtTE0NvJBLZ9eBuf+uep2412Bchjx3onOocnHr3ItSLb6+Y/udQ8Z2fLu9rkMqYXp1qb2/7w5u+1L2YqNVgX4S8vO5zK5vrKi48UvbO3pmPryyr/ez0+5ecmkC39p5aXypOIGoB9kXOK+pvW15XOTM8Ea377Yr6tasOfYVqMNf8Ypwx0evWrWci7UvdVoM1iJUqZufRXVZz8+ojW/nlzkx/hN/BgQF+32g9vu2+7Ro4sWOwL0IWdCFd2vlAUXkJF+PNbxfX3Ixa+1JfaqIWYF8EHqk5in0u79kr6Ab4sWOw98JnUTTdH72xbWfg6NI2C7DHAIPuZNt5LLOMu4F/LMBeQsBAO/HqP1Yf2WIIulaDvUR34sj7Y9/pXP3KFtZIXtLNjZaqBt9WfStfcZHbT6IErnUOjj585lNP3l2y/06j5KMK8If9F8rKyjbq8r8YJaO0zODBGKl/s6ThjpVP3Z22sOYCqgCju+xtgbFm/syvDnTFXYVDw0BuVQHu6+vV5j41UEypWMKhMbb/dHF16armzWayrQpw+8n2bdt1uNbMFFNSrhLdVQYyrwQwzvFoJKIhvsRAAbmwZJRDIxWfSgB3d3Xb6VWcBEH3Wue/cGgsrS41uV8qAcwAzPaWyc3QzNuVp97CoXFj2w5zHBrZazD7l0yhmT/LPpf94NC4+vRbODT0bw1lIfz0Gny2qxt0gbl0fo1Uf/8OPB5Z1JQfj4AuDg3QNc2hkb0GDwwMAK18oKInksjM3sCSl2ArHBphQRcxZrMfjAbj2DITA/+4Mtyhkb0G+yeyEFFmyTtSfwKHhpnuKhdJph+DQwSDT6yKQ2NpeQmLIp+q8I+sBTi9bEGX+Bsnuir9AyaVSA4wfZalHr+JrDbs2cNN+S2Ej5nuKheA4kBJMslia5PNL8abkPZZD7udafE3TtNm+qPMCeYjv3akWY5zujD2M/rtjv8u+Sm/cfcL8M/xV/oQBb/Gtl3AGnvyrAuH1zUYrb9ccdjD7m9J6ZTATUNNyYM4Y8GnL1j1RSDX3vnEXQ6X1/xqpOFU4Jo9MzQOG+5KvCSOy8mOgUtlh4Y3vEYjA29AIAzQcCTggh+GkQLTFyKBsOdUCp/ARFdzRyoeYJ6nX0R3ttGGAsQY2Gg4zU8FniEDs9gY+AQs936W0lWJZ878vTBvBzn3CA0mrmMHOiZPnDdkp4HFjsp2Vja+aG/Fagg10I186fWZyMTKpzcW33HTsppbmLMA6nR/ZLrnItIE2uWbOPW7OVz93gK80MHmlry/+2DFzrUz/x7Fes1GFpw8RWUlILqsrpJw9nBBKw2zAM9JwViHRu7mzfqil4Qo/iYLvAsd4HDF31iAM5PAXIKjh8+ser4uRBEaiS0kZ8/mezelyvhRuBrMTGp0/xnib244cFdm/WJxaWKKkS/3ELEc0SNsjRC2TGmS/k4e/8ETTzjPEpEOcW7K9h1VcBFLnEjIs91drx07+nJLS9JKCxRgr+JvAOBbTU0/fOYZLgguJpKJmDWEnmkGHcoTmMzj3z14kGt5HMAa9uzdtXv3uXd7BgYG0VTJncZNpwdIJCQlUx4bC9bfFkjteH/w8OHny732ui/f2/7Hk3F0dn71/pYXX3InToHW1487ZX787LPONTS/8dhj/EmZuz6/Qe7HXvPf2PLQ+f7jj3f9tStpjQWnwR7G34iexWVDwtJyU5JQYmyxrnJHrrHnSU03iuvcr6payMQzd95g/Tq5z8k/SbA1R7yxEW12DDJqjQlJdTSw4ACWfXJPYhm6u7u2L8515ZyRF+kj91ePHSOw/CfPPbdt+31YYJUTmpFItLb2Hh4fHBjkyK4ALBaYf/HLITH+VBwFCgtgHBqMvp6gi6BBrrR00YFpIARUxmBnouQgKhcyY+IL5KLTzgRKgETFy8pK0+YpBfhoVGkepwowNkHmh5lODs0pT8w6B4o8zH+TdGoDNnz7evuchsee2yM7NJ1AJlN0BblGy6UwBAGe+/JnZVWlY4eZZM3r8cJBNy4UJ+qqADOowJOKhTEH0VhOcGiM//xtNoI89CfLiZ7E9qLWzpwWbQbsVENv3LNMyF84fMgxy8yrMRJShtEXms4JXmy1YrYFVYCpgxypIT1j6NOBIlRzcDD+QN7cGqm1lbWNAIMd/t7Bg5Ig2h1mFriokKArr3MQQ8210CSBuNMhuBM3OqTUK8WlwvDw8No11fwqljenWNr4m6xZZX3CksZZxiAfvixsZI3EL3/K6oVirJ1633svVV37HnpIHpcvhaUkj0OQOyyEnGeho45FkoiOpEzQGFmcheuTNv4mx+aAXOxyNkdqio/HrYPdn1IFGKJpF++K/GkrhkOD+JvIpt/4V6Mok06MUWVHv1XapQowhsLFwqjUpLkM6OKr4ps2asmdMYwkAo31HMWVRyyOofa7jfQkjHlGA6USwLTB8Zn53QZP6KdFd+Cjj4BNBjx3/aMYhTFgiS5JT1j1m4jSLJr5MzPGEOUxHN13ivjIVUe2pkroy94LU1ZxLTHRdclOIUtPkgyJWz90HyWAmd9XVVVx1F9m7Xg88MgoLrT1S2Qu/uZ0/2x0crz5HFFzSRlgESlrevEqkAUsFZ/0A3rDRkqHNI2jionAjmH6paTM11U2TFQoe17m6vPnCBie+P0HRPAzvWKSxbV7LbRO9nZkkUNhuQ7dpDJpM5XG4MQnaXzsysxznLIjmBiVLni7HCCT9T2jLDVyLavY/IBWZKhkopOaL9O8WkkT+hKtQUQOcTnySpvED3syjv9I0uky4dDwxjltI1c2ADMlwUXnbFVq49WlIpeEvmBMwPqVAx2Jj+MOJN4lFk7Teq0Hss3UGGLHnHiDTJ/1qXxadxXDMIY67rgYZtmZWDiMicswo4WmT43yimxmGozuEhDEeomJNPEGHvSvnEmoJPTlUAIHFIihjHXWszr6xeGF89CyfctW3a7dD2Ko2cNR3P/JmX3fCaQEOOkCg70OgspkF/OF/0vHdx5TV6Ce0JddQhKpCCW66b49e4GQd7zKlvtIdIRf9meIv+HrvjIOsL2xVadaAcaxl+ToClKbW0qeOM+7FQ1PpUr8DSlfef9UWj6d95AZAk+ObDDniHzhKDG/JCJ3fztTvAbz5DBv7eqP8trMtFLLkcscH880oW9xzS051mjO4xgkAKJzA1aqBYJwuwhgTunQL3g1qodxLT4JJYv4GxPeQ+ahNMAY0wVYQMb7mlJRvm6i0dqhW3/tIQeWlE4JKCVhYRNGJTOPVzP47OgYkkQhO+a9fUol51WSSdb8O+fPYAEMeb1irB54deREp275UZcsH1QSoSVZJrFqVJmX+sG3O02Lbqx8eBtxRf8jadN0hOaEP32WGSMpX8t6rh+90t/JQldjZp6soJrn4YGioJoQVL3h0GBzXqgdFE5Z1xsCDTYzoW/WEtf8oOkA53H+Gz1IGw0wR044Lraquc7DA0V6xGpOLeYC7NOBInNEr4cTQwE29oXaelDxsBYTZ9HWoZHPALPngUODTRIcpR62s2BJmaXB6hEaBQtYpg03aAy26GYKnkp5gwC2Dg0VwDItYwrAmcbfZNrOgi1vBMD5ndA32L4VPMB5n9C3oAHOj4S+wULoXnuQyyTr0NDQMwIz0RZdDehSRTAabONv9KBLLZo0mM0DJlPSKht/ow1dfQDjYeYUtjTMw4S+OiUV0ro0abAcc5qD2dOEviEVuk629Y3Bl4p+tnxL9dTf/sPBJ14vPtXzSVH5DTZUw2+w9QF8+TMvzl4aL769YubjK7w4bumaUiLsCcfxu4UFTl8fwGOP/mlmaGLF1mrMddp4/AJHxcPm6wPYQ6YtKXUJaJpkqTNkS3orAQuwt/I0jpoF2DhIvGXIAuytPI2j9j/g0yYVnBgOdwAAAABJRU5ErkJggg==)
Answer:(i) Given: l1||l2
∠1 = (2x + 20)° and ∠2 = (3x-10)°
∵ Corresponding angles are equal
∴ ∠1 = ∠2
⇒ (2x + 20)° = (3x-10)°
⇒ x = 30°
ii. Given: l1||l2
∠1 = (2x)° and ∠2 = (3x + 20)°
∵ Co-interior angles are supplementary
∴ ∠1 + ∠2 = 180°
⇒ (2x)° + (3x + 20)° = 180°
⇒ 5x = 160°
⇒ x = 32°
Question 15.The angles of a triangle are in the ratio of 1:2:3. Find the measure of each angle of the triangle.
Answer:Let the angles of a triangle be x, 2x and 3x.
We know that sum of the angles of a triangle is 180°.
⇒ x + 2x + 3x = 180°
⇒ 6x = 180°
⇒ x = 30°
So, the angles of the triangle are 30°, 2×30° = 60° and 3× 30° = 90°.
Question 16.In ΔABC, ∠A + ∠B = 70° and ∠B + ∠C = 135°. Find the measure of each angle of the triangle.
Answer:Given ∠A + ∠B = 70° and ∠B + ∠C = 135°
We know that sum of the angles of a triangle is 180°.
∠A + ∠B + ∠C = 180° …I
Also, ∠A + ∠B + ∠B + ∠C = 70° + 135°
⇒ (∠A + ∠B + ∠C) + ∠B = 205°
From I,
⇒ 180° + ∠B = 205°
⇒ ∠B = 205° - 180°
⇒ ∠B = 25°
Putting in given equations,
∠A + ∠B = 70° and ∠B + ∠C = 135°
⇒ ∠A + 25° = 70° and 25° + ∠C = 135°
⇒ ∠A = 45° and ∠C = 110°
So, the angles of the triangle are 45°, 25° and 110°.
Question 17.In the given figure at right, side BC of ΔABC is produced to D. Find ∠A and ∠C.
![](data:image/png;base64,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)
Answer:Given ∠B = 40° and ∠ACD = 120°
∵ ∠ACD is an external angle
∴ ∠ACD = ∠A + ∠B
⇒ ∠A + 40° = 120°
⇒ ∠A = 80°
We know that sum of the angles of a triangle is 180°.
∠A + ∠B + ∠C = 180°
⇒ 80° + + 40° + ∠C = 180°
⇒ ∠C = 180° - 120°
⇒ ∠C = 60°
Find the complement of each of the following angles.
i. 63° ii. 24°
iii. 48° iv. 35°
v. 20°
Answer:
i. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 63° = 90° - 63° = 27°
ii. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 24° = 90° - 24° = 66°
iii. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 48° = 90° - 48° = 42°
iv. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 35° = 90° - 35° = 55°
v. We know that two angles are said to be complementary if their sum is 90°.
So, the complement of 20° = 90° - 20° = 70°
Question 2.
Find the supplement of each of the following angles.
i. 58° ii. 148°
iii. 120° iv. 40°
v. 100°
Answer:
i. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 58° = 180° - 58° = 122°
ii. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 148° = 180° - 148° = 32°
iii. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 120° = 180° - 120° = 60°
iv. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 40° = 180° - 40° = 140°
v. We know that two angles are said to be supplementary if their sum is 180°.
So, the supplement of 100° = 180° - 100° = 80°
Question 3.
Find the value of x in the following figures.
i.
ii.
Answer:
i. In the given figure,
AB is a straight line, hence all the angles formed on it are supplementary.
⇒ (x – 20)° + x° + 40° = 180°
⇒ 2x + 20° = 180°
⇒ 2x = 180° - 20°
⇒ 2x = 160°
⇒ x = 80°
ii. In the given figure,
AB is a straight line, hence all the angles formed on it are supplementary.
⇒ (x + 30)° + (115 – x)° + x° = 180°
⇒ x + 145° = 180°
⇒ x = 180° - 145°
⇒ x = 35°
Question 4.
Find the angles in each of the following.
The angle which is two times its complement.
Answer:
Let the complement of an angle be x°
According to the question,
Angle = 2x°
We know that two angles are said to be complementary if their sum is 90°.
So, 2x° + x° = 90°
⇒ 3x° = 90°
⇒ x° = 30°
∴ Angle = 2x° = 2× 30° = 60°, Compliment = x° = 30°
Question 5.
Find the angles in each of the following.
The angle which is four times its supplement.
Answer:
Let the supplement of an angle be x°
According to the question,
Angle = 4x°
We know that two angles are said to be supplementary if their sum is 180°.
So, 4x° + x° = 180°
⇒ 5x° = 180°
⇒ x° = 36°
∴ Angle = 4x° = 4× 36° = 144°, Compliment = x° = 36°
Question 6.
Find the angles in each of the following.
The angles whose supplement is four times its complement.
Answer:
Let the compliment of an angle be x°
According to the question,
Supplement of the angle = 4x°
We know that two angles are said to be supple mentary if their sum is 180°.
So, required angle = 180° - 4x° ..(I)
Also, if the two angles are complimentary their sum is 90°.
⇒ required angle = 90° - x° …(II)
Equating I and II,
90° - x° = 180° - 4x°
⇒ 3x° = 90°
⇒ x° = 30°
Hence, required angle = 90° - x° = 90° - 30° = 60°
Question 7.
Find the angles in each of the following.
The angle whose complement is one sixth of its supplement.
Answer:
Let the supplement of an angle be x°
According to the question,
Compliment of the angle
We know that two angles are said to be supple-mentary if their sum is 180°.
So, ..(I)
Also, if the two angles are complimentary their sum is 90°.
…(II)
Equating I and II,
⇒ x° = 108°
Hence, required angle = 180° - x° = 180° - 108° = 72°
Question 8.
Find the angles in each of the following.
Supplementary angles are in the ratio 4:5.
Answer:
Let the supplementary angles be 4x and 5x
We know that two angles are said to be supple mentary if their sum is 180°.
⇒ 4x + 5x = 180°
⇒ 9x = 180°
⇒ x = 20°
So, the supplementary angles will be 4× 20° = 80° and 5× 20° = 100°
Question 9.
Find the angles in each of the following.
Two complementary angles are in the ratio 3:2.
Answer:
Let the complimentary angles be 3x and 2x
We know that two angles are said to be complimentary if their sum is 90°.
⇒ 2x + 3x = 90°
⇒ 5x = 90°
⇒ x = 18°
So, the complimentary angles will be 2×18° = 36° and 3×18° = 54°
Question 10.
Find the values of x, y in the following figures.
Answer:
Given: ∠ADC = 3x and ∠BDC = 2x
Here AB is the straight line.
∴ ∠ADC and ∠BDC are linear pair.
⇒ ∠ADC + ∠BDC = 180°
⇒ 3x + 2x = 180°
⇒ 5x = 180°
⇒ x = 36°
Question 11.
Find the values of x, y in the following figures.
Answer:
Given: ∠ADC = (3x + 5)° and ∠BDC = (2x – 25)°
Here AB is the straight line.
∴ ∠ADC and ∠BDC are linear pair.
⇒ ∠ADC + ∠BDC = 180°
⇒ 3x + 5 + 2x - 25 = 180°
⇒ 5x - 20 = 180°
⇒ 5x = 200°
⇒ x = 40°
Question 12.
Find the values of x, y in the following figures.
Answer:
Given: ∠AOD = y° ,∠DOC = 90°,∠COB = x° ,∠AOE = 3x and ∠BOE = 60°
Here AB is the straight line.
∴ ∠AOE and ∠BDC are linear pair.
⇒ ∠ADC + ∠BOE = 180°
⇒ 3x + 60° = 180°
⇒ 3x = 120°
⇒ x = 40°
Also, ∠AOD, ∠DOC and ∠BOC are linear pair.
⇒ ∠AOD + ∠DOC + ∠BOC = 180°
⇒ y + 90° + x° = 180°
⇒ y = 90° - 40°
⇒ y = 50°
Question 13.
Let l1 || l2 and m1 is a transversal. If ∠F = 65°, find the measure of each of the remaining angles.
Answer:
Given: ∠F = 65°
∠B = ∠F = 65° {Corresponding angle}
∠H = ∠F = 65° {vertically opposite angle}
∠D = ∠B = 65° {vertically opposite angle}
Also, ∠C + ∠F = 180° {co-interior angles are supplementary}
⇒ ∠C = 180° - 65°
⇒ ∠C = 115°
∠E = ∠C = 115° {Alternate interior angle}
∠G = ∠C = 115° {Corresponding angle}
∠A = ∠G = 115° {Alternate exterior angle}
Question 14.
For what value of x will l1 and l2 be parallel lines.
i.
ii.
Answer:
(i) Given: l1||l2
∠1 = (2x + 20)° and ∠2 = (3x-10)°
∵ Corresponding angles are equal
∴ ∠1 = ∠2
⇒ (2x + 20)° = (3x-10)°
⇒ x = 30°
ii. Given: l1||l2
∠1 = (2x)° and ∠2 = (3x + 20)°
∵ Co-interior angles are supplementary
∴ ∠1 + ∠2 = 180°
⇒ (2x)° + (3x + 20)° = 180°
⇒ 5x = 160°
⇒ x = 32°
Question 15.
The angles of a triangle are in the ratio of 1:2:3. Find the measure of each angle of the triangle.
Answer:
Let the angles of a triangle be x, 2x and 3x.
We know that sum of the angles of a triangle is 180°.
⇒ x + 2x + 3x = 180°
⇒ 6x = 180°
⇒ x = 30°
So, the angles of the triangle are 30°, 2×30° = 60° and 3× 30° = 90°.
Question 16.
In ΔABC, ∠A + ∠B = 70° and ∠B + ∠C = 135°. Find the measure of each angle of the triangle.
Answer:
Given ∠A + ∠B = 70° and ∠B + ∠C = 135°
We know that sum of the angles of a triangle is 180°.
∠A + ∠B + ∠C = 180° …I
Also, ∠A + ∠B + ∠B + ∠C = 70° + 135°
⇒ (∠A + ∠B + ∠C) + ∠B = 205°
From I,
⇒ 180° + ∠B = 205°
⇒ ∠B = 205° - 180°
⇒ ∠B = 25°
Putting in given equations,
∠A + ∠B = 70° and ∠B + ∠C = 135°
⇒ ∠A + 25° = 70° and 25° + ∠C = 135°
⇒ ∠A = 45° and ∠C = 110°
So, the angles of the triangle are 45°, 25° and 110°.
Question 17.
In the given figure at right, side BC of ΔABC is produced to D. Find ∠A and ∠C.
Answer:
Given ∠B = 40° and ∠ACD = 120°
∵ ∠ACD is an external angle
∴ ∠ACD = ∠A + ∠B
⇒ ∠A + 40° = 120°
⇒ ∠A = 80°
We know that sum of the angles of a triangle is 180°.
∠A + ∠B + ∠C = 180°
⇒ 80° + + 40° + ∠C = 180°
⇒ ∠C = 180° - 120°
⇒ ∠C = 60°
Exercise 4.3
Question 1.If an angle is equal to one third of its supplement, its measure is equal to
A. 40°
B. 50°
C. 45°
D. 55°
Answer:Let the required angle be x and the supplement of x is y.
As we know if two angles are supplement of each other, then sum of those two angles is 180°.
⇒ x + y = 180
⇒ y = 180 – x
Now, according to question;
Required angle is equal to one third of its supplement;
i.e. ![](data:image/png;base64,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)
⇒ y = 3x
Now putting the value of y in this equation,
⇒ 180 – x = 3x
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
⇒ ![](data:image/png;base64,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)
Hence, required angle is 45°.
Hence, correct option is (c).
Question 2.In the given figure, OP bisect ∠ BOC and OQ bisect ∠ AOC. Then ∠ POQ is equal to
![](data:image/jpeg;base64,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)
A. 90°
B. 120°
C. 60°
D. 100°
Answer:Suppose that
BOC = 2x and ∠ AOC = 2y
Now, by question; OP and OQ is the bisect or of ∠ BOC and ∠ AOC respectively.
⇒ ![](data:image/png;base64,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)
Similarly,
![](data:image/png;base64,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)
Now,
∠ POQ = ∠ POC + ∠ COQ = x + y
So, we have to find ∠ POQ i.e.x + y Now, ∠ BOC + ∠ AOC = 180
⇒ 2x + 2y = 180
⇒ 2(x + y) = 180
⇒ x + y = 180/2
⇒ x + y = 90°
Hence, ∠ POQ = 90°.
Question 3.The complement of an angle exceeds the angle by 60°. Then the angle is equal to
A. 25°
B. 30°
C. 15°
D. 35°
Answer:Let the required angle be x and it’s complement is y.
So, ![](data:image/png;base64,iVBORw0KGgoAAAANSUhEUgAAAGkAAAAXCAMAAAA/dJG5AAAAAXNSR0IArs4c6QAAAHtQTFRFAAAAAAAAAAA6AABmADpmADqQAGa2OgAAOjpmOjqQOmY6OmaQOma2OpCQOpDbZgAAZgBmZjoAZjpmZmYAZrb/kDoAkDo6kGYAkGY6kLaQkLbbkNv/tmYAtpA6ttv/tv//25A627Zm27aQ2////7Zm/9uQ/9u2//+2///b60YTPwAAAAF0Uk5TAEDm2GYAAAAJcEhZcwAADsQAAA7EAZUrDhsAAAAZdEVYdFNvZnR3YXJlAE1pY3Jvc29mdCBPZmZpY2V/7TVxAAABY0lEQVRIS+1UXXfCIAwNaotunXUq+1C2MW2F//8LlwBnpZRW7c529mAeOD0hyU1uLgW42Y2B32Xg8MjY7J0wzAtjC/qquPckkI3YjuxHTXZg5PQDgUR2PAn80qs9fGY99UYj6XKJJe1ZTfZ4sC3UBXpWIBnLjoqhI7QIyQiKqjmj5EGrORWiccAOpsvczbQAkJjdGSF22FTX5bC5mSjcpegST78n8ii6HpgJQFGDKg9ikAxvLXw5xcbfkAGH5E5vNb9DIhtLliBWzNPZkZzgZq/IWQIJVJeUBJ05VEF757a19Ei0p2/T6w1RM8we0Sdbqulhz5axvXtFNItBTlrANrSrcl0+rON++gZz9RTpmVTuzUiSPoskkXhPMo7pJfDAC1IByS4UhMA3oksWlUkgVR2K01A1n+/czQn/S/eN8oZXG9xWocQvzro2EHd5kcSvrduNN6J4/pORUIqjCP/5iP+8wheT6Bxjai+xKQAAAABJRU5ErkJggg==)
⇒ y = 90 – x
Now, according to question;
Complement of the angle exceeds the angle by 60° i.e. y exceeds the x by 60°.
⇒ y – 60 = x
Now, putting the value of your in this equation,
⇒ (90 – x) – 60 = x
⇒ 30 – x = x
⇒ 2x = 30
⇒ x = 15°
Hence, required angle is 15°.
Question 4.Find the measure of an angle, if six times of its complement is 12° less than twice of its supplement.
A. 48°
B. 96°
C. 24°
D. 58°
Answer:Let the required angle be x and it’s complement is y and it’s supplement is z.
⇒ x + y = 90
i.e. y = 90 – x
And, x + z = 180
i.e. z = 180 – x
Now, according to question,
Six times of complement of x is 12°
less than twice of its supplement.
i.e. 6y = 2z – 12
Putting the value of y and z in this equation,
⇒ 6 (90 – x) = 2 (180 – x) – 12
⇒ 540 – 6x = 360 – 2x – 12
⇒ 6x – 2x = 540 – 360 + 12
⇒ 4x = 192
⇒ ![](data:image/png;base64,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)
⇒ x = 48°
Hence required angle is 48°.
Question 5.In the given figure, ∠ B:∠ C = 2:3, find ∠ B
![](data:image/jpeg;base64,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)
A. 120°
B. 52°
C. 78°
D. 130°
Answer:Let ∠ B and ∠ C be 2x and 3x respectively.
Now, As we know sum of two interior angles of a triangle is equal to third exterior angle;
⇒ ∠ B + ∠ C = 130°
⇒ 2x + 3x = 130°
⇒ 5x = 130°
⇒ x = 26°
So, ∠ B = 2x = 2×26° = 52°
Hence, required angle is 52°.
Question 6.ABCD is a parallelogram, E is the mid – point of AB and CE bisects ∠ BCD. Then ∠ DEC is
A. 60°
B. 90°
C. 100°
D. 120°
Answer:Given: ABCD is a parallelogram and E is the mid – point of AB such that CE bisects ∠ BCD.
construction: Join DE. The figure is given below:
![](data:image/jpeg;base64,/9j/4AAQSkZJRgABAQEAYABgAAD/2wBDAAoHBwgHBgoICAgLCgoLDhgQDg0NDh0VFhEYIx8lJCIfIiEmKzcvJik0KSEiMEExNDk7Pj4+JS5ESUM8SDc9Pjv/2wBDAQoLCw4NDhwQEBw7KCIoOzs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozs7Ozv/wAARCACmARwDASIAAhEBAxEB/8QAHwAAAQUBAQEBAQEAAAAAAAAAAAECAwQFBgcICQoL/8QAtRAAAgEDAwIEAwUFBAQAAAF9AQIDAAQRBRIhMUEGE1FhByJxFDKBkaEII0KxwRVS0fAkM2JyggkKFhcYGRolJicoKSo0NTY3ODk6Q0RFRkdISUpTVFVWV1hZWmNkZWZnaGlqc3R1dnd4eXqDhIWGh4iJipKTlJWWl5iZmqKjpKWmp6ipqrKztLW2t7i5usLDxMXGx8jJytLT1NXW19jZ2uHi4+Tl5ufo6erx8vP09fb3+Pn6/8QAHwEAAwEBAQEBAQEBAQAAAAAAAAECAwQFBgcICQoL/8QAtREAAgECBAQDBAcFBAQAAQJ3AAECAxEEBSExBhJBUQdhcRMiMoEIFEKRobHBCSMzUvAVYnLRChYkNOEl8RcYGRomJygpKjU2Nzg5OkNERUZHSElKU1RVVldYWVpjZGVmZ2hpanN0dXZ3eHl6goOEhYaHiImKkpOUlZaXmJmaoqOkpaanqKmqsrO0tba3uLm6wsPExcbHyMnK0tPU1dbX2Nna4uPk5ebn6Onq8vP09fb3+Pn6/9oADAMBAAIRAxEAPwD2aiiigAooooAKKKKACiiigAooooAKKKaHUuUDAsvJGeRQA6iiigAooqjcatb22qW2nuHMlwCQwHyr1xk++Dj6UAXqKzrTVnvblkjsZhCsjxmZnTGVJB43buo9Kstf2aXAtmu4FnJAERkAY56cdaALFFZ0+rNHqD2UNjNcPGis7K6KBuzj7zAnoajTXke+a3+yTiMXBt/OyhXfjPTO7HvigDVooooAKKilureAgTTxxk8gO4GfzqQEEAg5B6GnYBaKKKQBRRRQAUUUUAFFFFABRRRQAUVFHcwTMyxTRyMvUK4JFS0AFFFFABRRRQAUUUUAFFFFABRRRQAVgWcN9HqSalJAAt27JIoB3qh+5uHbG3/x41yXjrxfrWl+JHsrS8FlDBEjoNinzc9ScjkZ4wPSursPEGq3Gn2882gTiSSJWYCVByR6E5FdKpTjBS097zMHUjKVux0NFYv9taj/ANAGf/v8n+NH9taj/wBAGf8A7/J/jWfspf00X7SP9Jm1WFeeHGvJ7q7a6dbp5Ee3KuwSPZgplc4POSfrT/7a1H/oAz/9/k/xo/trUf8AoAz/APf5P8aPZS/poPaR/pMbp2j3FjfvMbXTmDzSSG4VCJsMScZx7461fk07zL4Xf2qZcEHywE2nH/Ac/rVL+2tR/wCgDP8A9/k/xo/trUf+gDP/AN/k/wAaPZS/poPaR/pMbe6NPLrMt/Ha6dciSONR9qQloypbocHru/SoU8OSjVnumSzCm8+0idUPnjgfJn07deh6VY/trUf+gDP/AN/k/wAaP7a1H/oAz/8Af5P8aPZS/poPaR/pM2qKxf7a1H/oAz/9/k/xrnfGXjHVdNsIY7e0bT5Z5Npmd0cquD0HOM9MmnGjNtLT70J1YpX/AEZ0WoRsNaEpaSNDbBd62xlBO48dDithDlFI6Edxj9K4bwZ4r1vUNNm+02Mmo+TLsW5RkTdxnB6Akeorof7a1H/oAz/9/k/xqp0535XbTzRMJx+JdfJm1RWL/bWo/wDQBn/7/J/jR/bWo/8AQBn/AO/yf41Hspf00X7SP9Jm1RWL/bWo/wDQBn/7/J/jR/bWo/8AQBn/AO/yf40eyl/TQe0j/SZtUVi/21qP/QBn/wC/yf40f21qP/QBn/7/ACf40eyl/TQe0j/SZtUVi/21qP8A0AZ/+/yf40f21qP/AEAZ/wDv8n+NHspf00HtI/0mbVFYv9taj/0AZ/8Av8n+NcB4h8deIrTxDPHG/wBiW2YBLJkVt/AxuPU7u2P51UMPObsrfeiZVoRV3+R22jW8yXNmSsh8tJFkV7cx+Xk8fNgbunv610dMhdpII3dCjsoLKf4TjpT6znPndy4R5VYKKKKgsKKKKACiiigAooooAKr397Dp1lLdznCRjOB1J7Ae5qxXPO/9uatv62Fg+E9Jph1P0X+daQjzO72RE5WWm4lhpizltR1S3ilvZ2D/ALxA3kj+FVz0x/OtWiiqk7u5KVlYKKKKQwooooAKKKKACkJCqWYgADJJPAqte6jb2AUSFnlfiOGMZdz7D+tVBYXOpkSaqQkOcrZxt8v/AAM/xH26UADahcaixi0kARg4e8kHyD/cH8R/SrFtpFnBHIskYuXmGJpJwHaT657e1XFVUUKqhVUYAAwBS0gMvw6qw6SLVFVBbSyRbVGAMMcfoa1KzdO/darqlv6yLMB/vLz+orSoAKKKKYBRRRQAUUUUAFFFFAFbUL1bC0adlLtkLHGOsjnooo0vRIbdFub2GGfUHbzJJ2QEqx7KewHQVX02P+19R/tRxm0tyUtFP8Z6NJ/QVvU5vlXL9/8AkKCu+b7gooorE1CiiigAooooAKKKKACiiq9/ew6dZS3c5wkYzgdSewHuaaTbshN2V2Z+u3sv7vS7Jtt1dg5cf8so/wCJ/wCg96mtbaKzto7aBdsca7VFVNKtZl82/vB/pl2Qzj/nmv8ACg+n860K3lZLlRkrt8zCiiioKCiiigAooqre6jb2Cr5pLSPxHEgy7n2FAFkkAEkgAckntWY2o3GoOYdJVSgOHvHHyL/uj+I/pSCxutTIk1T91B1Wzjbg/wC+e/0HFaiqqIERQqqMBQMACkBVstNgsi0gLS3D/wCsnkOXb/Aewq3RRTAKKKKAM1v3PiZD2ubUr+KNn+RrSrN1X91eaZc/3Ljy2Ps4I/nitKkAUUUUwCiiigAooooAKzdRaS/uk0e3YqZF33Mi/wDLOL0+rdPpmrOoXqWFo07KXbIWOMdZHPRRU2j6e9jbM9wwe7uG8y4f1b0HsOgqk+Vc33Ev3nyl2KKOCJIokCRooVVHQAU+iisDYKKKKACiiigAooooAKKKKACued/7c1bf1sLB8J6TTDqfcL/OrOuXkv7vS7Jtt1dA5cf8so/4n/oPepra2is7aO2gXbHGu1RW8Fyx5urMpPmduhLRRRUjCiiigAoJwCScAdSaq3uoW9gimZiXfiOJBl3PoBVQWV3qhD6n+5t+q2aN1/327/QcUgFfUp752g0lVYA4e7cfu0/3f7x/SrFlpsNkzS7mmuX+/PIcu3+A9hVpEWNFSNQiKMBVGABTqACiiimAUUUUAFFFFAGbr6k6NNIv3oSso/4CwNaKsHUOOjDIqO6hFxaTQnpJGy/mKraLMZ9GtHb73lBW+o4P8qQF6iiimAUUUUAFFFZuoNJf3SaPbsVMi77mRf8AlnF6fVun0zTirsTdkO02P+19Q/tNxm1tyUtFP8Z6NJ/QVu0yKKOCJIokCRooVVHQAU+onLmfkXGNkFFFFQUFFFFABRRRQAUUUUAFV7+9h06ylupzhIxnA6k9gPc1Yrnmf+3NW8zrYWD4T0mmHU+4X+daQjd3eyInKy03JNKtZl82/vB/pl2Qzj/nmv8ACg+n860KKKqTu7kpWVgooqte6hb2CBp2O5zhI0GXc+gHekMs+/YVlyanNeyNBpKLJg4e6f8A1SfT+8fpxSfY7vVTu1HNvbdVtEblv99h/IVpoiRRrHGioijCqowBSAq2WmQ2btMzNPdP9+4k5Y+w9B7CrlFFMAooooAKKKKACiiigAooooAB1rM0T93Fd2v/AD73TqPoTuH8606zbX914gv4u00ccw/VT/IUgNKiiimAUUUfpQBWv71bC0adlLtkLHGOsjnoo/GptH09rG2Z7hg93cN5lw/q3oPYdBVLTY/7W1D+1HGbW3JS0U/xHo0n9B7Vu05vlXL9/wDkKCu+YKKKKxNQooooAKKKKACiiigAooqvf3sOnWUt1OcJGM4HVj2A9zTSbdkJuyuzP1y8lPl6XZtturoHLj/llH/E39B71NbW0VnbR28C7Y412qKqaXazKJb68H+mXZDSD/nmv8KD6fzrQraVkuVGSu3zMKKrXt/b2EYed8FjhI1GXc+gHeqX2S81b5tQzbWva0Rvmf8A32H8hUlDpNTmvJGt9JRZSpw9y/8Aqo/p/ePsKnstMitHM7s1xdP9+eTlj7D0HsKtxxxwxrHEioijCqowBTqQBRRRTAKKKKACiiigAooooAKKKKACiiigArNuv3XiGxl7TRSQn6jDD+taVZmt/u47O6/54XSEn2Pyn+dIDTooPWimAVm6gXv7lNHt2K+Yu+6kX/lnF6fVun0zVm/vVsLRpipdyQscY6yOeiiptH09rG2Zp2D3dw3mTuO7eg9h0FUnyrm+4l+8+UuxRJDEkUShERQqqOgAp9FFYGwUUUUAFFFFABRRRQAUUUUAFc8z/wBt6t5nWwsXwnpLMOp9wv8AOrWuXkpMel2bbbq6B3OP+WUf8Tf0HvTJJrHRLGKInZGo2RRqMs59AO5raK5Y83VmUnzO3Qu1ly6pLdyNb6SizOpw9w/+qj/+KPsKb9lvNW+a/wB1raHpao3zP/vsP5CtOKKOGJYokWONRhVUYAqRlWy0uK0kNxI7XN24+aeTr9AP4R7CrtFFMAooooAKKKKACiiigAooooAKKKKACiiigAooooAKo61CZ9Fu0UZYRll+o5H8qdqWpR6dAreW000rbIYU+9I3p7D1PaslrK61BSdVu3ffn9xAxSJM9uOW+p/KkBuW1wk9vDIHXMiK2M88ipjx14+tcZ4f0vT7jRYfPsomljLRyMR8xKsRyaspaTy6qmi2F5OLZ033iO+8RR9grHkFumM9M1UVcUnY2tNj/ta//tRxm1gJS0U/xHo0n9B7Vu0yONIYkijUIiAKqjoAKfUTlzPyLjGyCiiioKCiiigAooooAKKKKAKsOpWk83kxzZfJUZUgEjqASMHoelJf6la6dbySzyqCi7gmfmb0AH1qjBa3TJb2z2zRiG6aYysykEbmIxg5ycj07151rPg3xFf+LLjNq8hnud8d6SNiJnIOc5GBxt9q6I04OWsrIwlOajornX2F1dTyTy2saz6hcnM87f6q3HZAf4iB2HetSy0uO1lNzK7XN2w+aeTr9FH8I9hSRaLqcESxRavHGijAVbNQB+tP/srV/wDoNL/4CL/jQ7N3cl+P+Q1dK1n+H+Zcoqn/AGVq/wD0Gl/8BF/xo/srV/8AoNL/AOAi/wCNK0f5l+P+QXl2/L/MuUVT/srV/wDoNL/4CL/jR/ZWr/8AQaX/AMBF/wAaLR/mX4/5BeXb8v8AMuUVT/srV/8AoNL/AOAi/wCNH9lav/0Gl/8AARf8aLR/mX4/5BeXb8v8y5RVP+ytX/6DS/8AgIv+NH9lav8A9Bpf/ARf8aLR/mX4/wCQXl2/L/MuUVT/ALK1f/oNL/4CL/jR/ZWr/wDQaX/wEX/Gi0f5l+P+QXl2/L/MuUVT/srV/wDoNL/4CL/jR/ZWr/8AQaX/AMBF/wAaLR/mX4/5BeXb8v8AMuUVT/srV/8AoNL/AOAi/wCNH9lav/0Gl/8AARf8aLR/mX4/5BeXb8v8y5RVP+ytX/6DS/8AgIv+NH9lav8A9Bpf/ARf8aLR/mX4/wCQXl2/L/MuUVT/ALK1f/oNL/4CL/jR/ZesD/mNr/4CL/jRaP8AMvx/yC8u35f5mZA327U7rUG5VHNvbj+6qn5iPq2fyFXKrweFdUto/Li1/am4sAbNT1JJ7+pp/wDwjms/9DCP/AJf8aLR/mX4/wCQ7vt+X+ZiWt8mlSavb7RJKt3uggU/NIZACAB9a6bTLB9C0W4uJcTX0itPcP8A33x0HsOgrzvSvB3iODxfbSTWj74LsSyX5I2uufmYHOTuHGO2favXSMjB5FOpywSjF3vq/wDImF5ttq39blO0EySRmW/E3mx7tjKo545XHbn37c1dqvb2FraOXghVGIxnrgeg9B7CrFYyab0NYppBRRRUlBRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFABRRRQAUUUUAFFFFAH//2Q==)
Let ∠ BCD = 2x
⇒ ∠ BCE = ∠ ECD = ∠ BCD/2![](data:image/png;base64,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)
Now, let ∠ ADC = 2y
And as we joined DE it will also bisect ∠ ADC.
Therefore, ∠ EDC = ![](data:image/png;base64,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)
Now, as we know sum of adjacent angles of parallelogram is equal to 180°.
⇒ ∠ BCD + ∠ ADC = 180°
⇒ 2x + 2y = 180°
⇒ 2 (x + y) = 180°
⇒ x + y = 90°
Now, In triangle CED,
∠ CED + ∠ EDC + ∠ DCE = 180°
⇒ ∠ CED + x + y = 180°
⇒ ∠ CED + 90° = 180°
⇒ ∠ CED = 180° – 90°
⇒ ∠ CED = 90°
Hence required angle is 90°.
If an angle is equal to one third of its supplement, its measure is equal to
A. 40°
B. 50°
C. 45°
D. 55°
Answer:
Let the required angle be x and the supplement of x is y.
As we know if two angles are supplement of each other, then sum of those two angles is 180°.
⇒ x + y = 180
⇒ y = 180 – x
Now, according to question;
Required angle is equal to one third of its supplement;
i.e.
⇒ y = 3x
Now putting the value of y in this equation,
⇒ 180 – x = 3x
⇒
⇒
⇒
Hence, required angle is 45°.
Hence, correct option is (c).
Question 2.
In the given figure, OP bisect ∠ BOC and OQ bisect ∠ AOC. Then ∠ POQ is equal to
A. 90°
B. 120°
C. 60°
D. 100°
Answer:
Suppose that
BOC = 2x and ∠ AOC = 2y
Now, by question; OP and OQ is the bisect or of ∠ BOC and ∠ AOC respectively.
⇒
Similarly,
Now,
∠ POQ = ∠ POC + ∠ COQ = x + y
So, we have to find ∠ POQ i.e.x + y Now, ∠ BOC + ∠ AOC = 180
⇒ 2x + 2y = 180
⇒ 2(x + y) = 180
⇒ x + y = 180/2
⇒ x + y = 90°
Hence, ∠ POQ = 90°.
Question 3.
The complement of an angle exceeds the angle by 60°. Then the angle is equal to
A. 25°
B. 30°
C. 15°
D. 35°
Answer:
Let the required angle be x and it’s complement is y.
So,
⇒ y = 90 – x
Now, according to question;
Complement of the angle exceeds the angle by 60° i.e. y exceeds the x by 60°.
⇒ y – 60 = x
Now, putting the value of your in this equation,
⇒ (90 – x) – 60 = x
⇒ 30 – x = x
⇒ 2x = 30
⇒ x = 15°
Hence, required angle is 15°.
Question 4.
Find the measure of an angle, if six times of its complement is 12° less than twice of its supplement.
A. 48°
B. 96°
C. 24°
D. 58°
Answer:
Let the required angle be x and it’s complement is y and it’s supplement is z.
⇒ x + y = 90
i.e. y = 90 – x
And, x + z = 180
i.e. z = 180 – x
Now, according to question,
Six times of complement of x is 12°
less than twice of its supplement.
i.e. 6y = 2z – 12
Putting the value of y and z in this equation,
⇒ 6 (90 – x) = 2 (180 – x) – 12
⇒ 540 – 6x = 360 – 2x – 12
⇒ 6x – 2x = 540 – 360 + 12
⇒ 4x = 192
⇒
⇒ x = 48°
Hence required angle is 48°.
Question 5.
In the given figure, ∠ B:∠ C = 2:3, find ∠ B
A. 120°
B. 52°
C. 78°
D. 130°
Answer:
Let ∠ B and ∠ C be 2x and 3x respectively.
Now, As we know sum of two interior angles of a triangle is equal to third exterior angle;
⇒ ∠ B + ∠ C = 130°
⇒ 2x + 3x = 130°
⇒ 5x = 130°
⇒ x = 26°
So, ∠ B = 2x = 2×26° = 52°
Hence, required angle is 52°.
Question 6.
ABCD is a parallelogram, E is the mid – point of AB and CE bisects ∠ BCD. Then ∠ DEC is
A. 60°
B. 90°
C. 100°
D. 120°
Answer:
Given: ABCD is a parallelogram and E is the mid – point of AB such that CE bisects ∠ BCD.
construction: Join DE. The figure is given below:
Let ∠ BCD = 2x
⇒ ∠ BCE = ∠ ECD = ∠ BCD/2
Now, let ∠ ADC = 2y
And as we joined DE it will also bisect ∠ ADC.
Therefore, ∠ EDC =
Now, as we know sum of adjacent angles of parallelogram is equal to 180°.
⇒ ∠ BCD + ∠ ADC = 180°
⇒ 2x + 2y = 180°
⇒ 2 (x + y) = 180°
⇒ x + y = 90°
Now, In triangle CED,
∠ CED + ∠ EDC + ∠ DCE = 180°
⇒ ∠ CED + x + y = 180°
⇒ ∠ CED + 90° = 180°
⇒ ∠ CED = 180° – 90°
⇒ ∠ CED = 90°
Hence required angle is 90°.