Page 1 Points to Remember : Angle. The figure formed by two rays with the same initial point, is called the angle. The common initial point is called the vertex of the angle and the rays forming the angle are called its arms or sides. In the figure, rays OA and OB with common initial point O form the angle AOB or angle BOA. We denote it as AOB or BOA or simply O. Interior And Exterior of An Angle Let AOB be an angle. Then, the set of all the points which lie inside the angle, form the interior of the angle. Any point on any of the arms of the angle is said to lie on the angle. The set of all those points which lie outside the angle, form the exterior of the angle. In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB. Sol. (i) In the given figure, three angles are formed. Q. 1. Name three examples of angles from your daily life. Sol. Three examples are : Tongs, Scissors and Compasses. Q. 2. Name the vertex and the arms of ABC, given in the figure below : Sol. In the given angle ABC, the vertex is B and arms are BA and BC . Q. 3. How many angles are formed in the figure given below ? Name them. Page 2 Points to Remember : Angle. The figure formed by two rays with the same initial point, is called the angle. The common initial point is called the vertex of the angle and the rays forming the angle are called its arms or sides. In the figure, rays OA and OB with common initial point O form the angle AOB or angle BOA. We denote it as AOB or BOA or simply O. Interior And Exterior of An Angle Let AOB be an angle. Then, the set of all the points which lie inside the angle, form the interior of the angle. Any point on any of the arms of the angle is said to lie on the angle. The set of all those points which lie outside the angle, form the exterior of the angle. In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB. Sol. (i) In the given figure, three angles are formed. Q. 1. Name three examples of angles from your daily life. Sol. Three examples are : Tongs, Scissors and Compasses. Q. 2. Name the vertex and the arms of ABC, given in the figure below : Sol. In the given angle ABC, the vertex is B and arms are BA and BC . Q. 3. How many angles are formed in the figure given below ? Name them. Names of the angles are : ABC, BAC and ACB (ii) In the given figure, four angles are formed. Names of the angles are : ABC, BCD, CDA and BAD (iii) In the given figure, eight angles are formed. Names of the angles are : ABC, BCD, CDA, BAD, ABD, DCB, ADBand BDC Q. 4. In Fig. list the points which : (i) are in the interior of AOB (ii) are in the exterior of AOB (iii) lie on AOB Sol. In the given figure : (i) Points S and Q are in the interior of AOB (ii) Points P and R are in the exterior of AOB. (iii) Points A, O, B, N, T lie on AOB. Q. 5. See the adjacent Fig. and state which of the following statements are true and which are false (i) Point C is in the interior of AOC (ii) Point C is in the interor of AOD (iii) Point D is in the interior of AOC (iv) Point B is in the exterior of AOD. (v) Point C lies on AOB. Sol. (i) False (ii) True (iii) False (iv) True (v) False Q. 6. In the adjoining figure write another name for : (i) 1 (ii) 2 (iii) 3 Sol. In the given figure, another name for : (i) 1 is EPB (ii) 2 is PQC Page 3 Points to Remember : Angle. The figure formed by two rays with the same initial point, is called the angle. The common initial point is called the vertex of the angle and the rays forming the angle are called its arms or sides. In the figure, rays OA and OB with common initial point O form the angle AOB or angle BOA. We denote it as AOB or BOA or simply O. Interior And Exterior of An Angle Let AOB be an angle. Then, the set of all the points which lie inside the angle, form the interior of the angle. Any point on any of the arms of the angle is said to lie on the angle. The set of all those points which lie outside the angle, form the exterior of the angle. In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB. Sol. (i) In the given figure, three angles are formed. Q. 1. Name three examples of angles from your daily life. Sol. Three examples are : Tongs, Scissors and Compasses. Q. 2. Name the vertex and the arms of ABC, given in the figure below : Sol. In the given angle ABC, the vertex is B and arms are BA and BC . Q. 3. How many angles are formed in the figure given below ? Name them. Names of the angles are : ABC, BAC and ACB (ii) In the given figure, four angles are formed. Names of the angles are : ABC, BCD, CDA and BAD (iii) In the given figure, eight angles are formed. Names of the angles are : ABC, BCD, CDA, BAD, ABD, DCB, ADBand BDC Q. 4. In Fig. list the points which : (i) are in the interior of AOB (ii) are in the exterior of AOB (iii) lie on AOB Sol. In the given figure : (i) Points S and Q are in the interior of AOB (ii) Points P and R are in the exterior of AOB. (iii) Points A, O, B, N, T lie on AOB. Q. 5. See the adjacent Fig. and state which of the following statements are true and which are false (i) Point C is in the interior of AOC (ii) Point C is in the interor of AOD (iii) Point D is in the interior of AOC (iv) Point B is in the exterior of AOD. (v) Point C lies on AOB. Sol. (i) False (ii) True (iii) False (iv) True (v) False Q. 6. In the adjoining figure write another name for : (i) 1 (ii) 2 (iii) 3 Sol. In the given figure, another name for : (i) 1 is EPB (ii) 2 is PQC (iii) 3 is FQD EXERCISE 13B Q. 1. State the type of each of the following angles : Sol. (i) Obtuse angle (ii) Right angle (iii) Straight line (iv) Reflex angle (v) Acute angle (vi) Complete angle Q. 2. Classify the angles whose magnitudes are given below : (i) 30º (ii) 91º (iii) 179º (iv) 90º (v) 181º (vi) 360º (vii) 128º (viii) (90·5)º (ix) (39·3)º (x) 80º (xi) 0º (xii) 15º Sol. We know that an acute angle is less than 90º (ii) a right angle is equal to 90º (iii) an obtuse angle is greater than 90º but less than 180º (iv) an angle equal to 180º is a straight angle (v) angle greater than 180º but less than 360º is called a reflex angle (vi) angle equal to 360º is called a complete angle and angle equal to 0º is called a zero angle. Now the angles are : (i) acute (ii) obtuse (iii) obtuse (iv) right (v) reflex (vi) complete (vii) obtuse (viii) obtuse (ix) acute (x) acute (xi) zero (xii) acute Ans. Q. 3. How many degrees are there in : (i) One right angle ? (ii) Two right anlges ? (iii) Three right angles ? (iv) Four right angles ? (v) 2 3 right angle ? (vi) 1½ right angle ? Sol. (i) One right angle = 90 (ii) Two right angles = (2 × 90) = 180 (iii) Three right angles = (3 × 90) = 270 (iv) Four right angles = (4 × 90) = 360 (v) 2 3 right angle F H G I K J 2 3 90 60 (vi) 1½ right angle F H G I K J 1 1 2 90 F H G I K J 3 2 90 135 Q. 4. How many degrees are there in the angle between the hour hand and the minute hand of a clock, when it is (i) 3 o clock (ii) 6 o clock (iii) 12 o clock (iv) 9 o clock Sol. (i) When it is 3 o clock, the minute hand is at 12, and hour hand is at 3 as O B A C O D F O E O Q P O G H O P (i) (ii) (iii) (iv) (v) (vi) Page 4 Points to Remember : Angle. The figure formed by two rays with the same initial point, is called the angle. The common initial point is called the vertex of the angle and the rays forming the angle are called its arms or sides. In the figure, rays OA and OB with common initial point O form the angle AOB or angle BOA. We denote it as AOB or BOA or simply O. Interior And Exterior of An Angle Let AOB be an angle. Then, the set of all the points which lie inside the angle, form the interior of the angle. Any point on any of the arms of the angle is said to lie on the angle. The set of all those points which lie outside the angle, form the exterior of the angle. In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB. Sol. (i) In the given figure, three angles are formed. Q. 1. Name three examples of angles from your daily life. Sol. Three examples are : Tongs, Scissors and Compasses. Q. 2. Name the vertex and the arms of ABC, given in the figure below : Sol. In the given angle ABC, the vertex is B and arms are BA and BC . Q. 3. How many angles are formed in the figure given below ? Name them. Names of the angles are : ABC, BAC and ACB (ii) In the given figure, four angles are formed. Names of the angles are : ABC, BCD, CDA and BAD (iii) In the given figure, eight angles are formed. Names of the angles are : ABC, BCD, CDA, BAD, ABD, DCB, ADBand BDC Q. 4. In Fig. list the points which : (i) are in the interior of AOB (ii) are in the exterior of AOB (iii) lie on AOB Sol. In the given figure : (i) Points S and Q are in the interior of AOB (ii) Points P and R are in the exterior of AOB. (iii) Points A, O, B, N, T lie on AOB. Q. 5. See the adjacent Fig. and state which of the following statements are true and which are false (i) Point C is in the interior of AOC (ii) Point C is in the interor of AOD (iii) Point D is in the interior of AOC (iv) Point B is in the exterior of AOD. (v) Point C lies on AOB. Sol. (i) False (ii) True (iii) False (iv) True (v) False Q. 6. In the adjoining figure write another name for : (i) 1 (ii) 2 (iii) 3 Sol. In the given figure, another name for : (i) 1 is EPB (ii) 2 is PQC (iii) 3 is FQD EXERCISE 13B Q. 1. State the type of each of the following angles : Sol. (i) Obtuse angle (ii) Right angle (iii) Straight line (iv) Reflex angle (v) Acute angle (vi) Complete angle Q. 2. Classify the angles whose magnitudes are given below : (i) 30º (ii) 91º (iii) 179º (iv) 90º (v) 181º (vi) 360º (vii) 128º (viii) (90·5)º (ix) (39·3)º (x) 80º (xi) 0º (xii) 15º Sol. We know that an acute angle is less than 90º (ii) a right angle is equal to 90º (iii) an obtuse angle is greater than 90º but less than 180º (iv) an angle equal to 180º is a straight angle (v) angle greater than 180º but less than 360º is called a reflex angle (vi) angle equal to 360º is called a complete angle and angle equal to 0º is called a zero angle. Now the angles are : (i) acute (ii) obtuse (iii) obtuse (iv) right (v) reflex (vi) complete (vii) obtuse (viii) obtuse (ix) acute (x) acute (xi) zero (xii) acute Ans. Q. 3. How many degrees are there in : (i) One right angle ? (ii) Two right anlges ? (iii) Three right angles ? (iv) Four right angles ? (v) 2 3 right angle ? (vi) 1½ right angle ? Sol. (i) One right angle = 90 (ii) Two right angles = (2 × 90) = 180 (iii) Three right angles = (3 × 90) = 270 (iv) Four right angles = (4 × 90) = 360 (v) 2 3 right angle F H G I K J 2 3 90 60 (vi) 1½ right angle F H G I K J 1 1 2 90 F H G I K J 3 2 90 135 Q. 4. How many degrees are there in the angle between the hour hand and the minute hand of a clock, when it is (i) 3 o clock (ii) 6 o clock (iii) 12 o clock (iv) 9 o clock Sol. (i) When it is 3 o clock, the minute hand is at 12, and hour hand is at 3 as O B A C O D F O E O Q P O G H O P (i) (ii) (iii) (iv) (v) (vi) shown in the figure, clearly, the angle between the two hands = 90 . (ii) When it is 6 o clock, the minute hand is at 12 and the hour hand is at 6 as shown in the figure. Clearly, the angle between the two hands of the clock is a straight angle is i.e. 180º. (iii) When it is 12 o clock, both the hands of the clock lie at 12 as shown in the figure. Clearly, the angle between the two hands = 0 . (iv) When it is 9 o clock, the minute hand is at 12 and the hour hand is at 9 as shown in the figure. Clearly, the angle between the two hands = 90º. Q. 5. Using a rular, draw an acute angle, an obtuse angle and a straight angle. Sol. (i) Take the rular and draw any ray OA. Again using the rular, starting from O, draw a ray OB in such a way that the angle formed is less than 90 . Then, AOB is the required acute angle. (ii) Take the rular and draw any ray OA. Now, starting from O, draw another ray OB, with the help of the rular, such that the angle formed is greater than a right angle. Then, AOB is the required obtuse angle. (iii) Take a rular and draw any ray OA. Now, starting from O, draw ray OB in the opposite direction of the ray OA. Then AOB is the required straight angle. Page 5 Points to Remember : Angle. The figure formed by two rays with the same initial point, is called the angle. The common initial point is called the vertex of the angle and the rays forming the angle are called its arms or sides. In the figure, rays OA and OB with common initial point O form the angle AOB or angle BOA. We denote it as AOB or BOA or simply O. Interior And Exterior of An Angle Let AOB be an angle. Then, the set of all the points which lie inside the angle, form the interior of the angle. Any point on any of the arms of the angle is said to lie on the angle. The set of all those points which lie outside the angle, form the exterior of the angle. In the figure, P lies in the interior of AOB, Q lies on AOB and R lies in the exterior of AOB. Sol. (i) In the given figure, three angles are formed. Q. 1. Name three examples of angles from your daily life. Sol. Three examples are : Tongs, Scissors and Compasses. Q. 2. Name the vertex and the arms of ABC, given in the figure below : Sol. In the given angle ABC, the vertex is B and arms are BA and BC . Q. 3. How many angles are formed in the figure given below ? Name them. Names of the angles are : ABC, BAC and ACB (ii) In the given figure, four angles are formed. Names of the angles are : ABC, BCD, CDA and BAD (iii) In the given figure, eight angles are formed. Names of the angles are : ABC, BCD, CDA, BAD, ABD, DCB, ADBand BDC Q. 4. In Fig. list the points which : (i) are in the interior of AOB (ii) are in the exterior of AOB (iii) lie on AOB Sol. In the given figure : (i) Points S and Q are in the interior of AOB (ii) Points P and R are in the exterior of AOB. (iii) Points A, O, B, N, T lie on AOB. Q. 5. See the adjacent Fig. and state which of the following statements are true and which are false (i) Point C is in the interior of AOC (ii) Point C is in the interor of AOD (iii) Point D is in the interior of AOC (iv) Point B is in the exterior of AOD. (v) Point C lies on AOB. Sol. (i) False (ii) True (iii) False (iv) True (v) False Q. 6. In the adjoining figure write another name for : (i) 1 (ii) 2 (iii) 3 Sol. In the given figure, another name for : (i) 1 is EPB (ii) 2 is PQC (iii) 3 is FQD EXERCISE 13B Q. 1. State the type of each of the following angles : Sol. (i) Obtuse angle (ii) Right angle (iii) Straight line (iv) Reflex angle (v) Acute angle (vi) Complete angle Q. 2. Classify the angles whose magnitudes are given below : (i) 30º (ii) 91º (iii) 179º (iv) 90º (v) 181º (vi) 360º (vii) 128º (viii) (90·5)º (ix) (39·3)º (x) 80º (xi) 0º (xii) 15º Sol. We know that an acute angle is less than 90º (ii) a right angle is equal to 90º (iii) an obtuse angle is greater than 90º but less than 180º (iv) an angle equal to 180º is a straight angle (v) angle greater than 180º but less than 360º is called a reflex angle (vi) angle equal to 360º is called a complete angle and angle equal to 0º is called a zero angle. Now the angles are : (i) acute (ii) obtuse (iii) obtuse (iv) right (v) reflex (vi) complete (vii) obtuse (viii) obtuse (ix) acute (x) acute (xi) zero (xii) acute Ans. Q. 3. How many degrees are there in : (i) One right angle ? (ii) Two right anlges ? (iii) Three right angles ? (iv) Four right angles ? (v) 2 3 right angle ? (vi) 1½ right angle ? Sol. (i) One right angle = 90 (ii) Two right angles = (2 × 90) = 180 (iii) Three right angles = (3 × 90) = 270 (iv) Four right angles = (4 × 90) = 360 (v) 2 3 right angle F H G I K J 2 3 90 60 (vi) 1½ right angle F H G I K J 1 1 2 90 F H G I K J 3 2 90 135 Q. 4. How many degrees are there in the angle between the hour hand and the minute hand of a clock, when it is (i) 3 o clock (ii) 6 o clock (iii) 12 o clock (iv) 9 o clock Sol. (i) When it is 3 o clock, the minute hand is at 12, and hour hand is at 3 as O B A C O D F O E O Q P O G H O P (i) (ii) (iii) (iv) (v) (vi) shown in the figure, clearly, the angle between the two hands = 90 . (ii) When it is 6 o clock, the minute hand is at 12 and the hour hand is at 6 as shown in the figure. Clearly, the angle between the two hands of the clock is a straight angle is i.e. 180º. (iii) When it is 12 o clock, both the hands of the clock lie at 12 as shown in the figure. Clearly, the angle between the two hands = 0 . (iv) When it is 9 o clock, the minute hand is at 12 and the hour hand is at 9 as shown in the figure. Clearly, the angle between the two hands = 90º. Q. 5. Using a rular, draw an acute angle, an obtuse angle and a straight angle. Sol. (i) Take the rular and draw any ray OA. Again using the rular, starting from O, draw a ray OB in such a way that the angle formed is less than 90 . Then, AOB is the required acute angle. (ii) Take the rular and draw any ray OA. Now, starting from O, draw another ray OB, with the help of the rular, such that the angle formed is greater than a right angle. Then, AOB is the required obtuse angle. (iii) Take a rular and draw any ray OA. Now, starting from O, draw ray OB in the opposite direction of the ray OA. Then AOB is the required straight angle. EXERCISE 13 C Q. 1. Measure each of the following angles with the help of a protractor and write the measures in degrees. Sol. (i) Place the protractor in such a way that its centre is exactly at the vertex O of the given angle AOB and the base line lies along the arm OA. Read off the mark through which the arm OB passes, starting from 0 on the side A. We find that AOB = 45 . (ii) The given angle is PQR. Place the protractor in such a way that its centre is exactly on the vertex Q of the given angle and the base line lies along the arm QR.Read off the mark through which the arm QP passes, starting from 0 on the side of R. We find that PQR = 67 . (iii) The given angle is DEF. Place the protractor in such a way that its centre is exactly on the vertex E of the given angle and the base line lies along the arm ED. Read off the mark through which the arm EF passes, starting from 0 on the side of D. We find that the DEF = 130°. (iv) The given angle is LMN. Place the protractor in such a way that its centre is exactly on the vertex M of the given angle and the base line lies along the arm ML. Read off the mark through which the arm MN passes, starting from 0 on the side of L. We find that the LMN = 50°. (v) The given angle is RST. Place the protractor in such a way that its centre is exactly on the vertex S of the given angle and the base line lies along the arm SR. Read off the mark through which the arm ST passes, starting from 0° on the side of R.Read More

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