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Angles, lines, rays, and segments

Angles, lines, rays, and segments. MATHEMATICS. Contents. Recap the terms. Test Yourself - 1. Angles in daily life. Congruent angles. Pairs of angles: Types. What is an angle?. Test Yourself - 2. Naming an angle. Pairs of angles formed by a transversal.

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Angles, lines, rays, and segments

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  1. Angles, lines, rays, and segments MATHEMATICS

  2. Contents • Recap the terms • Test Yourself - 1 • Angles in daily life • Congruent angles • Pairs of angles: Types • What is an angle? • Test Yourself - 2 • Naming an angle • Pairs of angles formed by a transversal • Interior and exterior of an angle • Test Yourself - 3 • Measurement of angle • Types of angle: Right angle • Obtuse angle • Acute angle • Straight angle

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

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

  5. A B C Ray BAand rayBCare called thearmsof ABC. What Is An Angle ? When two non-collinear rays join with a common endpoint (origin) an angle is formed. Ray BA B Common endpoint Ray BC Common endpoint is called the vertex of the angle.Bis thevertexof ÐABC. Ray BA and BC are two non-collinear rays

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

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

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

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

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

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

  12. Examples Of Right Angle

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

  14. Examples Of Obtuse Angle

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

  16. Examples Of Acute Angle

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

  18. Examples Of Straight Angle

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

  20. D 300 E F D A 300 300 E F B C Congruent Angles Two angles that have the same measure are called congruent angles. Congruent angles have the same size and shape.

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

  22. A D A C D Common ray B E F B C Common vertex Adjacent Angles Two angles that have acommon vertexand acommon rayare called adjacent angles. Adjacent AnglesABDand DBC ABCand DEF are not adjacent angles Adjacent angles do not overlap each other.

  23. A C D B Vertically Opposite Angles Vertically opposite angles are pairs of angles formed bytwo lines intersecting at a point. ÐAPC = ÐBPD ÐAPB = ÐCPD P Four anglesare formed at the point of intersection. Vertically opposite angles arecongruent. Point of intersection ‘P’ is thecommon vertex of the four angles.

  24. A D 600 300 B E C F Complimentary Angles If the sum of two angles is 900, then they are called complimentary angles. ÐABCandÐDEFare complimentary because ÐABC + ÐDEF 600 + 300 = 900

  25. D p 700 300 Q E R F Contd…. If the sum of two angles is more than 900 or less than 900, then they not complimentary angles. ÐDEFandÐPQRare not complimentary because ÐDEF+ÐPQR 700 + 300 = 1000

  26. A P 1000 800 B Q R C Supplementary Angles If the sum of two angles is 1800 then they are called supplementary angles. ÐPQRandÐABCare supplementary, because ÐPQR + ÐABC 1000 + 800 = 1800

  27. D A 1100 800 E B F C Contd…. If the sum of two angles ismore than 1800orless than 1800, then they arenot supplementary angles. ÐDEFandÐPQRarenot supplementarybecause ÐABC + ÐDEF 1100 + 800 = 1900

  28. A 1200 600 D C Linear Pair Of Angles Two adjacent supplementary angles are called linear pair of angles. P ÐAPC + ÐAPD 600 + 1200 = 1800

  29. A D A 300 D 300 900 600 900 600 E C B E C B Name the adjacent angles and linear pair of angles in the given figure: Adjacent angles: ÐABD and ÐDBC ÐABE and ÐDBA Test Yourself 2 Linear pair of angles: ÐEBA, ÐABC ÐEBD, ÐDBC

  30. C A P D B Name the vertically opposite angles and adjacent angles in the given figure: Vertically opposite angles: ÐAPC and ÐBPD Adjacent angles:ÐAPC and ÐCPD ÐAPB and ÐCPD ÐAPB and ÐBPD

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

  32. Pairs Of Angles Formed by a Transversal • Corresponding angles • Alternate angles • Interior angles

  33. L Line L G Line M P A B Line N Q D E F Corresponding Angles When two parallel lines are cut by a transversal, pairs of corresponding angles are formed. ÐGPB = ÐPQE ÐGPA = ÐPQD ÐBPQ = ÐEQF ÐAPQ = ÐDQF Four pairs of corresponding angles are formed. Corresponding pairs of angles are congruent.

  34. L Line L G Line M P A B Line N Q D E F Alternate Angles Alternate angles are formed on opposite sides of the transversal and at different intersecting points. ÐBPQ = ÐDQP ÐAPQ = ÐEQP Two pairs of alternate angles are formed. Pairs of alternate angles are congruent.

  35. L Line L G Line M P A B Line N Q D E F Interior Angles The angles that lie in the area between the two parallel lines that are cut by a transversal, are calledinterior angles. ÐBPQ + ÐEQP = 1800 600 1200 1200 600 ÐAPQ + ÐDQP = 1800 The measures of interior angles in each pairadd up to 1800. A pair of interior angles lie on thesame sideof the transversal.

  36. Line L G Line M P A B 500 Line L G 1300 Line M Line N A P B Q E D Line L G F Line N Q D E Line M P A B F Line N Q E D F Name the pairs of the following angles formed by a transversal. Test Yourself 3 Corresponding angles Alternate angles Interior angles

  37. The End

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