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## Different Types of Angle

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**a**a a a a a b b b b b b c c c c c c a and b, b and c, c and a Angles at a Point C O O A B are three pairs of adjacent angles with a common vertex O. a, b and c are called angles at a point.**a**b c ∵ round angle = 360O [Abbreviation:∠s at a pt.] The sum of all angles at a point is 360O. C C A A B B ∴ a + b + c = 360O**a**210O = 60O Let us try the following question. Find a in the figure. Solution a + 90O + 210O = 360O (∠s at a pt.) a + 300O = 360O a= 360O- 300O**(a)**y y + 10O 170O y= 90O Follow-up question 2 Find the unknowns in the following figures. Solution (a) y + 10O + y + 170O= 360O (∠s at a pt.) 2y + 180O= 360O 2y = 180O**(b)**5x 5x 2x x= 30O Follow-up question 2 (cont’d) Solution (b) 5x + 5x + 2x= 360O (∠s at a pt.) 12x = 360O**Example 3**Solution**Example 4**Solution**B**a common arm common vertex b O O a and b They have a common arm OB and a common vertex O. Adjacent Angles on a Straight Line A B C O are two different angles. They are on the opposite sides of the common arm OB. a and b are called adjacent angles.**A**A B B a b b a C A C O and OC C O O are on the same straight line, If OA then a and b are called adjacent angles on a straight line.**B**b a A A C C O O ∵∠AOC = 180O ∴ a + b = 180O The sum of the adjacent angles on a straight line is 180O. [Abbreviation: adj. ∠s on st. line]**Let us try the following question.**O X Y 49O a R = 131O In the figure, XOY is a straight line, find a. ∵ XOY is a straight line. ∴ ∠XOY = 180O a + 49O = 180O (adj. ∠s on st. line) a = 180O- 49O**= 68O**Follow-up question 1 In each of the following figures, XOY is a straight line. Find the unknown. (a) 32O X m O Y Solution (a) ∵ XOY is a straight line. ∴∠XOY = 180O m + 32O + 90O = 180O (adj. ∠s on st. line) m = 180O- 122O**(b)**S R b b Y b O X b= 60O Follow-up question 1 (cont’d) Solution (b) ∵ XOY is a straight line. ∴ ∠XOY = 180O b + b + b = 180O (adj. ∠s on st. line) 3b= 180O**Example 1**Solution**Example 2**Solution**ID1 (P.157)**ID2 (P.158) ID4 (P.160) ID3 (P.159) 45o 170o 70o ID6 (P.162) ID7 (P.163) ID5 (P.160) 2 20o 30o 35o No.**point O.**are two straight lines intersecting at and CD AB A c D a b O d C B Vertically Opposite Angles a and b are opposite to each other without any common arm. They are called vertically opposite angles. Similarly, c and d are another pair of vertically opposite angles.**i.e. a = b and c = d**[Abbreviation: vert. opp. ∠s] A D c a b O C d B When two straight lines intersect, the vertically opposite angles formed are equal.**C**E 100O 44O A B h k F D h= 44O (vert. opp. ∠s) k= 100O (vert. opp. ∠s) Let us try the following question. In the figure, AB, CD and EF are straight lines. Find h and k. Solution**A**X C 45O O z 120O D B z= 75O (a) (vert. opp. ∠s) z + 45O = 120O Follow-up question 3 In the following figures, AOB, COD and EOF are straight lines. Find the unknowns. (a) Solution**F**A c 3d D 7d C 80O E B (b) c = 80O (vert. opp. ∠s) d= 10O Follow-up question 3(cont’d) (b) Solution 7d + c + 3d = 180O (adj. ∠s on st. line) 10d + 80O = 180O 10d = 100O**(vert. opp. s)**(vert. opp. s) Example 5 Solution**Example 6**Solution**Example 7**In the figure, AB, CD and EF are straight lines intersecting at O. Find ∠BOD and ∠COE. Solution**CP (P.164)**70o 120o 45o 60o No.**CP (P.165)**85o 40o**Angles Formed by a Transversal and Two Lines**In the figure, straight line EF intersects 2 straight lines AB and CD. E E A B D C F F EF is called the transversal of AB and CD.**a**e Corresponding Angles Consider the following figure. E A B D C F a and e are on the same side of AB and CD. (i.e. above the lines AB and CD). They are also on the same side of EF (i.e. on the right of EF). a and e are called a pair of corresponding angles.**Can you find any other pairs of corresponding angles in the**figure? b b c c d d f f g g h h E a A B e D C F b and f, Yes, they are c and g, d and h.**c**e c lies on the left of EF and e lies on the right of EF. c lies below AB and e lies above CD. Alternate Angles Consider the following figure. E A B D C F c and e lie between AB and CD. They are on the opposite sides of EF. c and e are called a pair of alternate angles.**Can you find another pair of alternate angles in the figure?**d d f f c e E b a A B D C g h F Yes, d and f.**c**f c lies below AB and f lies above CD. Both c and f lie on the left of EF. Interior Angles on the Same Side Consider the following figure. E A B D C F c and f lie between AB and CD. They are on the same side of EF. c and f are called a pair of interior angles on the same side.**c**d d d e e e f E b a A B D C g h F Yes, d and e. Can you find another pair of interior angles on the same side?**d**f E b a A B c D C e g E h a b F B A If AB and CD are parallel, can you see any relationships among the angles as shown? c d e f D C g h F Angles Formed by a Transversal and Parallel Lines**Corresponding Angles**E a B A e D C F If AB // CD, then a = e. [Abbreviation: corr. ∠s, AB // CD]**A**140O B C 40O a D b Then, a = b= and Consider the following example.In the figure, AB // CD. (corr. ∠s, CD // AB) 140O 40O (corr. ∠s, CD // AB)**Alternate Angles**E B A c e D C F If AB // CD, then c = e. [Abbreviation: alt. ∠s, AB // CD]**Then,**= a b= and Consider the following example. In the figure, AB // CD. B 115O A a 65O D b C (alt. ∠s, CD // AB) 115O 65O (alt. ∠s, CD // AB)**Interior Angles on the Same Side**E B A d e D C F If AB // CD, then d+ e = 180O. [Abbreviation: int. ∠s, AB // CD]**A**B 105O a C D Consider the following example. In the figure, AB // CD. (int. ∠s, CD // AB) Then, a + 105O = 180O a= 75O**H**F x B Q R A D P S C 84O E G AQE = CPE = 84O (corr. ∠s, AB // CD) (corr. ∠s, GH // CD) ARG = AQE = 84O (vert. opp. ∠s) x = ARG = 84O Follow-up question 4 Find the unknown in the following figure. (a) Solution (a)**B**D 5y 7y A C y= 15O Follow-up question 4 (cont’d) Find the unknown in the following figure. (b) Solution (b) 5y + 7y = 180O(int. ∠s, CD // AB) 12y = 180O**Example 8**Find x and y in the figure. Solution**Example 9**Find x and y in the figure. Solution**Example 10**Find t in the figure. Solution**Example 11**Find a and b in the figure. Solution**Example 12**In the figure, AB // DC // FE and ED // CBG. Find ∠ABG. Solution**Example 13**In the figure, AB // CD and EF // GH. Find x and y. Solution