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Class 8: Lines and Angles (Lecture Notes) – Part 2 Angles: an angle is the figure formed by two rays, called the sides of the angle, sharing a common endpoint, called the vertex of the angle. Two rays ⃗⃗⃗⃗⃗ 𝑂𝐴 and ⃗⃗⃗⃗⃗ 𝑂𝐵 having a common point will form an angle AOB which is written as ∠AOB. OA and OB are called the arms of the angle and O is called the vertex of the angle ∠AOB. Measure of an Angle: It is the amount of rotation through which one arm of the angle has to be rotated, about the vertex, to bring it to the position of the other arm. Angle is measured in degrees, denoted by °. A complete rotation around a point makes an angle of 360°. One degree (1°) = 60 minutes (also written as 60´). One Minute (1´) = 60 seconds (also written as 60´´). To draw angles, the commonly used equipment is called protector. Kinds of Angles Name of the Angle Description Acute Angle An angle whose measure is more than 0° but less than 90° is called an acute angle. Diagram 1 For more information please go to: https://icsemath.com/ Right Angle An angle whose measure is equal to 90° is called a right angle. Obtuse Angle An angle whose measure is more than 90° but less than 180° is called an obtuse angle. Straight Angle An angle whose measure is equal to 180° is called a straight angle. Reflex Angle An angle whose measure is more than 180° but less than 360° is called a reflex angle. Complete Angle An angle whose measure is equal to 360° is called a complete angle. Equal Angles: Two angles are said to be equal if they have the same measure. Bisector of an Angle: Any ray is called a bisector of an angle if ∠AOC = ∠COB 2 For more information please go to: https://icsemath.com/ Complimentary Angles: If the sum of two angles is 90°, then the angles are called complimentary angles. We can also say that Complementary angles are angle pairs whose measures sum to one right angle. Supplementary Angles: If the sum of two angles is 180°, then the angles are called supplementary angles. If the two supplementary angles are adjacent their non-shared sides form a straight line. Adjacent Angles: If two angles share one common arm and a common vertex in such a way that the other angle arms are on either side of the common arm then they are called adjacent angles. In this example we see that O is the common vertex, and OC is the common arm. Hence we can say that ∠AOC & ∠COB are adjacent angles. Linear Pair of Angles: If the adjacent angles are such that the, the non-common arms form a straight angle, then the angles are called linear pair of angles. In this case ∠AOB + ∠COB = 180° = ∠AOC Another way of looking at this is that is the sum of two adjacent angles is 180°, then they will form a linear pair of angles. One more important result that you should know is that the sum of angles around a point (or dot) is 360°. Vertically Opposite Angles: When two straight lines intersect at one point, they will form vertically opposite angles which are equal. As you see, lines AB and CD intersect at point O. It forms two pairs of vertically opposite angles, which are: ∠AOD & ∠COB are vertically opposite ∠AOC & ∠DOB are vertically opposite We can also prove that these angles are equal to each other. Given: Line AB and CD intersect at point O To Prove: i) ∠BOC = ∠AOD and ii) ∠AOC = ∠DOB 3 For more information please go to: https://icsemath.com/ Proof: Since ray OC stands on a straight line ∠AOC + ∠BOC = 180° [Linear Pair Axiom] Similarly, since ray OA stands on line CD ∠AOC + ∠AOD = 180° [Linear Pair Axiom] Therefore ∠AOC + ∠BOC = ∠AOC + ∠AOD Or ∠BOC = ∠AOD. Hence proved Similarly, you can prove ∠AOC = ∠DOB Perpendicular Lines: A line is said to be perpendicular to another line if the two lines intersect at a right angle. If AB and CD are two perpendicular lines, then they are denoted as AB ⊥ CD. 4 For more information please go to: https://icsemath.com/