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Chapter 1-4 ANGLES

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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: Right angle Types of angle: Right angle Obtuse angle Acute angle Straight angle Test Yourself - 1 Congruent angles Pairs of angles: Types Test Yourself - 2 Pairs of angles formed by a transversal Pairs of angles formed by a transversal Test Yourself - 3

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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 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: ABC or CBA We may also name this angle only by the single letter of the vertex, for example 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 A B C A B C BA C Types Of Angles

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Right angle: An angle whose measure is 90 degrees. Right AngleAcute AngleStraight AngleObtuse Angle

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Examples Of Right Angle

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Obtuse angle: An angle whose measure is greater than 90 degrees. Right AngleAcute AngleStraight AngleObtuse Angle

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Examples Of Obtuse Angle

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Acute angle: An angle whose measure is less than 90 degrees. Right AngleAcute AngleStraight AngleObtuse Angle

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Examples Of Acute Angle

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Straight angle: An angle whose measure is 180 degrees. Right AngleAcute AngleStraight AngleObtuse Angle

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Examples Of Straight Angle

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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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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 D E F Congruent Angles

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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. APC = BPD APB = 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 30 0 D E F ABC and DEF are complimentary because 60 0 + 30 0 = 90 0 ABC + DEF Complimentary Angles

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

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If the sum of two angles is 180 0 then they are called supplementary angles. PQR and ABC are supplementary, because 100 0 + 80 0 = 180 0 R Q P A B C 100 0 80 0 PQR + 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. DEF and PQR are not supplementary because ABC + 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 APC + APD Linear Pair Of Angles

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Name the adjacent angles and linear pair of angles in the given figure: Adjacent angles: ABD and DBC ABE and DBA Linear pair of angles: EBA, ABC C D B A E 60 0 30 0 90 0 EBD, DBC 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: APC and BPD APB and CPD Adjacent angles: APC and CPD APB and 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. GPB = PQE GPA = PQD BPQ = EQF APQ = 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 BPQ = DQP APQ = 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 60 0 BPQ + EQP = 180 0 APQ + DQP = 180 0

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Name the pairs of the following angles formed by a transversal. Line M B A Line N DE P Q G F Line L Line M B A Line N D E P Q G F Line L Line M B A Line N D E P Q G F Line L 50 0 130 0

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