**Exercise 8.1**

**Question 1: Write the complement of each of the following angles:**

**(i) 20°**

**(ii) 35°**

**(iii) 90°**

**(iv) 77°**

**(v) 30°**

**Solution:**

**Question 2 : Write the supplement of each of the following angles:**

**(i) 54°**

**(ii) 132°**

**(iii) 138°**

**Solution:**

**Question 3: If an angle is 28° less than its complement, find its measure?**

**Solution:**

**Question 4: If an angle is 30° more than one half of its complement, find the measure of the angle?**

**Solution:**

**Question 5: Two supplementary angles are in the ratio 4:5. Find the angles?**

**Solution:**

**Question 6: Two supplementary angles differ by 48°. Find the angles?**

**Solution:**

**Question 7: An angle is equal to 8 times its complement. Determine its measure?**

**Solution:**

**Question 8: If the angle (2x-10)° and (x-5)° are complementary angles. Find the value of x°.**

**Solution:**

**Question 9: If the complement of an angle is equal to supplement of thrice of it find the measure of it.**

**Solution:**

**Question 10: If an angle differ by is complementary by 10°. Find the angle.**

**Solution:**

**Question 11: If the supplement of an angle is three times its complement. Find the angle.**

**Solution:**

**Question 12: If the supplement of an angle is two third of its Determine the angle and its supplement.**

**Solution:**

**Question 13: An angle is 14° more than its complementary angle. What is its measure?**

**Solution:**

**Question 14: The measure of an angle is twice of its supplement angle. Find its measure.**

**Solution:**

**Exercise 8.2**

**Question 1: In the below Fig. OA and OB are opposite rays:**

**(i) If x = 25°, what is the value of y?**

**(ii) If y = 35°, what is the value of x?**

**Solution:**

**Question 2: In the below figure, write all pairs of adjacent angles and all the linear pairs.**

**Solution:**

**Question 3: In the given figure, find x. Further find ∠BOC, ∠COD and ∠AOD.**

**Solution:**

**Question 4: In figure, rays OA, OB, OC, OD and OE have the common end point 0. Show that ∠AOB + ∠BOC + ∠COD + ∠DOE + ∠EOA = 360°.**

**Solution:**

**Question 5: In figure, ∠AOC and ∠BOC form a linear pair. If a – 2b = 30°, find a and b?**

**Solution:**

**Question 6: How many pairs of adjacent angles are formed when two lines intersect at a point?**

**Solution:**

**Question 7: How many pairs of adjacent angles, in all, can you name in figure given?**

**Solution:**

**Question 8: In figure, determine the value of x.**

**Solution:**

**Question 9: In figure, AOC is a line, find x.**

**Solution:**

**Question 10: In figure, POS is a line, find x.**

**Solution:**

**Question 11: In the below figure, ACB is a line such that ∠DCA = 5x and ∠DCB = 4x. Find the value of x?**

**Solution:**

**Question 12: In the given figure, Given ∠POR = 3x and ∠QOR = 2x + 10, Find the value of x for which POQ will be a line?**

**Solution:**

**Question 13: In Fig: a is greater than b by one third of a right angle. Find the value of a and b?**

**Solution:**

**Question 14: What value of y would make AOB a line in the below figure, If ∠AOB = 4y and ∠BOC = (6y + 30) ?**

**Solution:**

**Question 15: If the figure below forms a linear pair, forms a linear pair ∠EOB = ∠FOC = 90° and ∠DOC = ∠FOG = ∠AOB = 30°. Find the measure of ∠FOE, ∠COB and ∠DOE**

**Name all the right angles**

**Name three pairs of adjacent complementary angles**

**Name three pairs of adjacent supplementary angles**

**Name three pairs of adjacent angles**

**Solution:**

**Question 16: In below fig. OP, OQ, OR and OS are four rays. Prove that: ∠POQ + ∠QOR + ∠SOR + ∠POS = 360°.**

**Solution:**

**Question: 17: In below fig, ray OS stand on a line POQ. Ray OR and ray OT are angle bisectors of ∠POS and ∠SOQ respectively. If ∠POS = x, find ∠ROT?**

**Solution:**

**Question 18: In the below fig, lines PQ and RS intersect each other at point O. If ∠POR: ∠ROQ = 5: 7. Find all the angles.**

**Solution:**

**Question 19: In the below fig. POQ is a line. Ray OR is perpendicular to line PQ. OS is another ray lying between rays OP and OR. Prove that ∠ROS = 1/2(∠QOS − ∠POS).**

**Solution:**

**Exercise 8.3**

**Question 1: In figure, lines l**

_{1}, and l_{2}intersect at O, forming angles as shown in the figure. If x = 45. Find the values of y, z and u.**Solution:**

**Question 2: In figure, three coplanar lines intersect at a point O, forming angles as shown in the figure. Find the values of x, y, z and u.**

**Solution:**

**Question 3: In figure, find the values of x, y and z.**

**Solution:**

**Question 4: In figure, find the value of x.**

**Solution:**

**Question 5: Prove that bisectors of a pair of vertically opposite angles are in the same straight line.**

**Solution:**

**Question 6: If two straight lines intersect each other, prove that the ray opposite to the bisector of one of the angles thus formed bisects the vertically opposite angle.**

**Solution:**

**Question 7: If one of the angles formed by intersecting lines is an angle then show that the each of the angles is a right angle.**

**Solution:**

**Question: 8: In the below fig. rays AB and CD intersect at O.**

**(i) Determine y when x = 60**

**(ii) Determine x when y = 40**

**Solution:**

**Question 9: In the below fig. lines AB. CD and EF intersect at O. Find the measures of ∠AOC, ∠COF, ∠DOE and ∠BOF.**

**Solution:**

**Solution:**

**Question 10: AB, CD and EF are three concurrent lines passing through the point O such that OF bisects BOD. If BOF = 35. Find BOC and AOD.**

**Solution:**

**Question 11: In below figure, lines AB and CD intersect at O. If ∠AOC + ∠BOE = 70° and ∠BOD = 40°, find ∠BOE and reflex ∠COE?**

**Solution:**

**Question 12: Which of the following statements are true (T) and which are false (F)?**

**(i) Angles forming a linear pair are supplementary.**

**(ii) If two adjacent angles are equal and then each angle measures 90**

**(iii) Angles forming a linear pair can both acute angles.**

**(iv) If angles forming a linear pair are equal, then each of the angles have a measure of 90**

**Solution:**

**Question 13: Fill in blanks so as to make the following statements true:**

**(i) If one angle of a linear pair is acute then its other angle will be______**

**(ii) A ray stands on a line, then the sum of the two adjacent angles so formed is ______**

**(iii) If the sum of two adjacent angles is 180, then the ______ arms of the two angles are opposite rays.**

**Solution:**

**Obtuse angle.**

**180°**

**Uncommon**arms of the two angles are opposite rays.

**Exercise 8.4**

**Question 1: In figure, AB, CD and ∠1 and ∠2 are in the ratio 3 : 2. Determine all angles from 1 to 8.**

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**Question 2: In figure, I, m and n are parallel lines intersected by transversal p at X, Y and Z respectively. Find ∠1, ∠2 and ∠3.**

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**Question 3: In figure, AB || CD || EF and GH || KL. Find ∠HKL.**

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**Question 4: In figure, show that AB || EF.**

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**Question 5 : In figure, if AB || CD and CD || EF, find ∠ACE.**

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**Question 6: In figure, PQ || AB and PR || BC. If ∠QPR = 102°, determine ∠ABC. Give reasons.**

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**Question 7: In figure, state which lines are parallel and why?**

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**Question 8: In figure, if l||m, n || p and ∠1 = 85°, find ∠2.**

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**Question 9: If two straight lines are perpendicular to the same line, prove that they are parallel to each other.**

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**Question 10: Prove that if the two arms of an angle are perpendicular to the two arms of another angle, then the angles are either equal or supplementary.**

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**Question: 11: In the below fig, lines AB and CD are parallel and P is any point as shown in the figure. Show that ∠ABP + ∠CDP = ∠DPB.**

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**Question: 12: In the below fig, AB ∥ CD and P is any point shown in the figure. Prove that: ∠ABP + ∠BPD + ∠CDP = 360°**

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**Question: 13: Two unequal angles of a parallelogram are in the ratio 2: 3. Find all its angles in degrees.**

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**Question: 14: If each of the two lines is perpendicular to the same line, what kind of lines are they to each other?**

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**Question: 15: In the below fig, ∠1 = 60° and ∠2 = 2/3rd of a right angle. Prove that l ∥ m.**

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**Question: 16: In the below fig, if l ∥ m ∥ n and ∠1 = 60°. Find ∠2.**

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**Question: 17: Prove that the straight lines perpendicular to the same straight line are parallel to one another.**

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**Question: 18: The opposite sides of a quadrilateral are parallel. If one angle of the quadrilateral is 60°. Find the other angles.**

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**Question: 19: Two lines AB and CD intersect at O. If ∠AOC + ∠COB + ∠BOD = 270°, find the measures of ∠AOC, ∠COB,**

**∠BOD, ∠DOA**

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**Question: 20: In the below figure, p is a transversal to lines m and n, ∠2 = 120° and ∠5 = 60°. Prove that m|| n.**

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**Question: 21: In the below fig. transversal t intersects two lines m and n, ∠4 = 110° and ∠7 = 65° is m ∥ n?**

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**Question: 22: Which pair of lines in the below fig. is parallel? Give reasons.**

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**Question: 23: If I, m, n are three lines such that I || m and n perpendicular to l, prove that n perpendicular to m.**

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**Question: 24: In the below fig, arms BA and BC of ∠ABC are respectively parallel to arms ED and EF of ∠DEF. Prove that ∠ABC = ∠DEF.**

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**Question: 25: In the below fig, arms BA and BC of ABC are respectively parallel to arms ED and EF of DEF Prove that**

**∠ABC + ∠DEP = 180°**

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**Question: 26: With of the following statements are true (T) and which are false (F)? Give reasons.**

**(i) If two lines are intersected by a transversal, then corresponding angles are equal.**

**(ii) If two parallel lines are intersected by a transversal, then alternate interior angles are equal.**

**(ii) Two lines perpendicular to the same line are perpendicular to each other.**

**(iv) Two lines parallel to the same line are parallel to each other.**

**(v) If two parallel lines are intersected by a transversal, then the interior angles on the same side of the transversal are equal.**

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**Question: 27: Fill in the blanks in each of the following to make the statement true:**

**(i) If two parallel lines are intersected by a transversal, then each pair of corresponding angles are ____________**

**(ii) If two parallel lines are intersected by a transversal, then interior angles on the same side of the transversal are _____________**

**(iii) Two lines perpendicular to the same line are _______ to each other**

**(iv) Two lines parallel to the same line are __________ to each other.**

**(v) If a transversal intersects a pair of lines in such a way that a pair of alternate angles we equal. then the lines are ___________**

**(vi) If a transversal intersects a pair of lines in such a way that the sum of interior angles on the seine side of transversal is 180'. then the lines are _____________**

**Solution:**

**Exercise VSAQs.......................**

**Question 1: Define complementary angles.**

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**Question 2: Define supplementary angles.**

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**Question 3: Define adjacent angles.**

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**Question 4: The complement of an acute angle is _____.**

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**Question 5: The supplement of an acute angle is _____.**

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**Question 6: The supplement of a right angle is _____.**

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