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STUDY NOTES FOR SSC CGL AND CPO www.mahendras.org myshop.mahendras.org Write us : content@mahendras.org www.mahendraguru.com STUDY NOTES FOR SSC CGL AND CPO FUNDAMENTAL CONCEPTS OF In the figure above, the angle is represented as ∠AOB. OA and GEOMETRY OB are the arms of ∠AOB. Point O is the vertex of ∠AOB. Point: The amount of turning from one arm (OA) to other (OB) is It is an exact location. It is a fine dot which has neither length called the measure of the angle ( AOB). nor breadth nor thickness but has position i.e., it has no Right angle: magnitude. An angle whose measure is 90 is called a right angle. Line segment: The straight path joining two points A and B is called a line segment points and a definite length. Ray: A line segment which can be extended in only one direction is called a ray. Acute angle: In angle whose measure is less than one right angle (i.e., less Intersecting lines: than 90), is called an acute angle. Two lines having a common point are called intersecting lines. The common point is known as the point of intersection. Obtuse angle: Concurrent lines: An angle whose measure is more than one right angle and less If two or more lines intersect at the same point, then they are than two right angles (i.e., less than 180 and more than 90) is known as concurrent lines. called an obtuse angle. Angles: When two straight lines meet at a point they form an angle Reflex angle: An angle whose measure is more than 180 and less than 360 is called a reflex angle. www.mahendras.org myshop.mahendras.org Write us : content@mahendras.org www.mahendraguru.com STUDY NOTES FOR SSC CGL AND CPO Complementary angles: If the sum of the two angles is one right angle (i.e., 90), they are called Complementary angles. In the above figure, ∠1 and ∠3 and angles ∠2 and ∠4 are vertically opposite angles. Therefore, the complement of an angle θ is equal to 90° − θ. Note: Vertically opposite angles are always equal. Bisector of an angle: If a ray or a straight line passing through the vertex of that angle, divides the angle into two angles of equal measurement, then that line is known as the Bisector of that angle. Supplementary angles: Two angles are said to be supplementary, if the sum of their measures is 180. Example: Angles measuring 130 and 50 are supplementary angles. Two supplementary angles are the supplement of each other. A point on an angle is equidistant from both the arms. Therefore, the supplement of an angle θ is equal to 180° − θ. Vertically opposite angles: When two straight lines intersect In the figure above, Q and R are the feet of perpendiculars each other at a point, the pairs of opposite angles so formed drawn from P to OB and OA. It follows that PQ = PR. are called vertically opposite angles Parallel lines: Two lines are parallel if they are coplanar and they do not intersect each other even if they are extended on either side. Transversal: A transversal is a line that intersects (or cuts) two or more coplanar lines at distinct points. www.mahendras.org myshop.mahendras.org Write us : content@mahendras.org www.mahendraguru.com STUDY NOTES FOR SSC CGL AND CPO Answer: As 67° + 113° = 180°, lines P and S, R and S, and S and U are parallel. Therefore, lines P, R, S and U are parallel to each other. Similarly, lines Q and T are parallel to each other. Example- In the figure given below, PQ and RS are two parallel lines and AB is a transversal. AC and BC are angle bisectors of ∠BAQ and ∠ABS, respectively. If ∠BAC = 30°, find ∠ABC and ∠ACB. In the above figure, a transversal t is intersecting two parallel lines, l and m, at A and B, respectively. Angles formed by a transversal of two parallel lines: Answer: ∠BAQ + ∠ABS = 180° [Supplementary angles] BAQ ABS 180 00 2 2 2 90 BACABC90 Therefore, ∠ABC = 60° and ∠ACB = 90°. Example- In the above figure, l and m are two parallel lines intersected by a transversal PS. The following properties of the angles can For what values of x in the figure given below are the lines be observed: P-A-Q and R–B-S parallel, given that AD and BD intersect ∠3 = ∠5 and ∠4 = ∠6 [Alternate angles] at D? ∠1 = ∠5, ∠2 = ∠6, ∠4 = ∠8, ∠3 = ∠7 [Corresponding angles] ∠4 + ∠5 = ∠3 + ∠6 = 180° [Supplementary angles] In the figure given below, which of the lines are parallel to each other? Answer: We draw a line DE, parallel to RS, as shown in the figure below: www.mahendras.org myshop.mahendras.org Write us : content@mahendras.org www.mahendraguru.com
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