LINES AND ANGLES

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DEEPAK SHARMA KAPIL GYANPEETH MATHEMATICS

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Contents Recap the terms Angles in daily life What is an angle ? Naming an angle Interior and exterior of an angle Measurement of angle Types of angle : Pairs of angles Congruent angles Test Yourself - 2 Pairs of angles formed by a transversal Test Yourself - 3 Examples of angles Tes t Yourself - 1

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Point An exact location on a plane is called a point. Line Line segment Ray A straight path on a plane, extending in both directions with no endpoints, is called a line. A part of a line that has two endpoints and thus has a definite length is called a line segment. A line segment extended indefinitely in one direction is called a ray. Recap Geometrical Terms

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If we look around us, we will see angles everywhere. Angles In Daily Life

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Common endpoint B C B A Ray BC Ray BA Ray BA and BC are two non-collinear rays When two non-collinear rays join with a common endpoint (origin) an angle is formed. What Is An Angle ? Common endpoint is called the vertex of the angle. B is the vertex of Ð ABC. Ray BA and ray BC are called the arms of  ABC.

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Fact: We can also think of an angle formed by rotating one ray away from its initial position.

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To name an angle, we name any point on one ray, then the vertex, and then any point on the other ray. For example: Ð ABC or Ð CBA We may also name this angle only by the single letter of the vertex, for example Ð B. A B C ` Naming An Angle

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An angle divides the points on the plane into three regions: A B C F R P T X Interior And Exterior Of An Angle Points lying on the angle (An angle) Points within the angle (Its interior portion. ) Points outside the angle (Its exterior portion. )

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Angles are accurately measured in degrees. Protractor is used to measure and draw angles. Measurement Of An Angle

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There are four main types of angles. Straight angle Right angle Acute angle Obtuse angle A B C B A C A B C A B C Types Of Angles

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Right Angle Obtuse Angle Acute Angle Straight Angle Obtuse Angle

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Right Angle Right angle: An angle whose measure is 90 degrees. Obtuse Angle Acute Angle Straight Angle Obtuse Angle

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Obtuse Angle Acute Angle Obtuse angle: An angle whose measure is greater than 90 degrees. Straight Angle Right Angle

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Acute Angle Straight Angle Acute angle: An angle whose measure is less than 90 degrees. Right Angle Obtuse Angle

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Straight angle: An angle whose measure is 180 degrees. Straight Angle Right Angle Acute Angle Obtuse Angle

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Examples Of Right Angle

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Examples Of Obtuse Angle

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Examples Of Acute Angle

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Examples Of Straight Angle

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Two angles that have the same measure are called congruent angles. Congruent angles have the same size and shape. A B C 30 0 D E F 30 0 D E F 30 0 Congruent Angles

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Test Yourself 1 A B C D E F P Q R Which of the angles below is a right angle, less than a right angle and greater than a right angle? Right angle Greater than a right angle Less than a right angle 1. 2. 3.

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Right Angle or Not? Right Angle Right Angle Not! Not! Not! Not!

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Right, Acute or Obtuse? Right Angle Right Angle Obtuse Greater Acute Less Acute Less Obtuse Greater

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Pairs Of Angles : Types Adjacent angles Vertically opposite angles Complimentary angles Supplementary angles Linear pairs of angles

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Adjacent Angles Two angles that have a common vertex and a common ray are called adjacent angles. C D B A Common ray Common vertex Adjacent Angles  ABD and  DBC Adjacent angles do not overlap each other. D E F A B C  ABC and  DEF are not adjacent angles

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Vertically Opposite Angles Vertically opposite angles are pairs of angles formed by two lines intersecting at a point. Ð APC = Ð BPD Ð APB = Ð CPD A D B C P Four angles are formed at the point of intersection. Point of intersection ‘P’ is the common vertex of the four angles. Vertically opposite angles are congruent.

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If the sum of two angles is 90 0 , then they are called complimentary angles. 60 0 A B C E 30 0 D F Ð ABC and Ð DEF are complimentary because 60 0 + 30 0 = 90 0 Ð ABC + Ð DEF Complimentary Angles

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70 0 D E F Q 30 0 p R If the sum of two angles is more than 90 0 or less than 90 0 , then they are not complimentary angles. Ð DEF and Ð PQR are not complimentary because 70 0 + 30 0 = 100 0 Ð DEF + Ð PQR Contd….

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If the sum of two angles is 180 0 then they are called supplementary angles. Ð PQR and Ð ABC are supplementary, because 100 0 + 80 0 = 180 0 Q R P A B C 100 0 80 0 Ð PQR + Ð ABC Supplementary Angles

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If the sum of two angles is more than 180 0 or less than 180 0 , then they are not supplementary angles. Ð DEF and Ð PQR are not supplementary because Ð ABC + Ð DEF 110 0 + 80 0 = 190 0 D E F 80 0 C B A 110 0 Contd….

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Two adjacent supplementary angles are called linear pair of angles. A 60 0 120 0 P C D 60 0 + 120 0 = 180 0 Ð APC + Ð APD Linear Pair Of Angles

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Name the adjacent angles and linear pair of angles in the given figure: Adjacent angles: Ð ABD and Ð DBC Ð ABE and Ð DBA Linear pair of angles: Ð EBA , Ð ABC C D B A E 60 0 30 0 90 0 Ð EBD , Ð DBC Test Yourself 2 C D B A E 60 0 30 0 90 0

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Name the vertically opposite angles and adjacent angles in the given figure: A D B C P Vertically opposite angles: Ð APC and Ð BPD Ð APB and Ð CPD Adjacent angles: Ð APC and Ð CPD Ð APB and Ð BPD

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A line that intersects two or more lines at different points is called a transversal . Line L (transversal) B A Line M Line N D C P Q G F Pairs Of Angles Formed by a Transversal Line M and line N are parallel lines. Line L intersects line M and line N at point P and Q. Four angles are formed at point P and another four at point Q by the transversal L. Eight angles are formed in all by the transversal L.

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Pairs Of Angles Formed by a Transversal Corresponding angles Alternate angles Interior angles

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Corresponding Angles When two parallel lines are cut by a transversal , pairs of corresponding angles are formed. Four pairs of corresponding angles are formed. Corresponding pairs of angles are congruent . Ð GPB = Ð PQE Ð GPA = Ð PQD Ð BPQ = Ð EQF Ð APQ = Ð DQF Line M B A Line N D E L P Q G F Line L

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Alternate Angles Alternate angles are formed on opposite sides of the transversal and at different intersecting points . Line M B A Line N D E L P Q G F Line L Ð BPQ = Ð DQP Ð APQ = Ð EQP Pairs of alternate angles are congruent . Two pairs of alternate angles are formed.

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The angles that lie in the area between the two parallel lines that are cut by a transversal, are called interior angles. A pair of interior angles lie on the same side of the transversal. The measures of interior angles in each pair add up to 180 0 . Interior Angles Line M B A Line N D E L P Q G F Line L 60 0 120 0 120 0 60 0 Ð BPQ + Ð EQP = 180 0 Ð APQ + Ð DQP = 180 0

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Name the pairs of the following angles formed by a transversal. Line M B A Line N D E P Q G F Line L Corresponding angles Line M B A Line N D E P Q G F Line L Alternate angles Line M B A Line N D E P Q G F Line L 50 0 130 0 Interior angles Test Yourself 3

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Review State whether the following are acute, right, or obtuse . 1. 2. 3. 4. 5. ? ? acute obtuse right obtuse acute

Complementary and Supplementary:

Complementary and Supplementary 1. Two angles are complementary. One measures 65 degrees. 2. Two angles are supplementary. One measures 140 degrees . Find the missing angle. Answer : 25 Answer : 40

Complementary and Supplementary:

Complementary and Supplementary Find the missing angle. You do not have a protractor. Use the clues in the pictures. 1. 2. x 55 165 x X =35 X=15

Vertical Angles:

Vertical Angles Find the missing angle. You do not have a protractor. Use the clues in the pictures. 58 x X=58

More drawings:

More drawings 20 C J D E F G H 70 90 70 20 90

Final Drawing:

Final Drawing 52 B A F E D C 60 G 68 68 60 52

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The End

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