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IRELATED ANGLES, , Line Segment: A line segment is a portion of a line. It has two end points. A Line Segment, line of segment AB is generally denoted by the symbol AB. A, Line: We cannot draw a line on a piece of paper. We can only show a portion, , ”, i, a, , oO, , of line on a paper. If we draw a portion of line and put arrow marks on, its two ends, we get a line. We can extend the line’endlessly in both the, directions. A line has no end points. A line POs is denoted by ‘the ore PO., , , , , , , , , , Angle: An angle is formed when two ines lin, point. This common pointis called the vertex of the ans, are called its arms or sides. In the figure,, the line segment OB and OA are arms, , ay, O, >, >, o, , FEVO ee eee oe, \, ow, , , , , , , , called an obtuse angle. For example, the angles, 100°, 120°, 150°, 175° etc. are all obtuse angles, , , , , , Right Angle: An angle whose measure is 90° is called'a right angle., , 90° (Right Angle), B, Complementary Angles: Two angles whose sum is 90° forma pair of complementary angles. Each angle, , oO, , of this pair is said to be the complement of the other angle. In the following diagram the “60°” is the, , complement of the “30°” angle and vice versa,, , Scanned with CamScanner
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i Supplementary Angles: Two angles whose sum is 180° form a pair of supplementary angles. Each, Ss es This pair ie eld fo be the supplement of the other angles. In the following figure, the “gg ie, supplement of the “100°” angle and vice-versa., , Q 100° Kor hae, —, , = Adjacent Angles: Two angles in a plane are said t be adjacent angles if they have a common vertex and, a common arm but no common interior points. SS, , the, , , , , , ? : oa ei Oe eB, , In the figure, two angles ZADC and ZBOC have a comm, , common interior points. Such Pairs of angles are calle, B)= Linear Pair: Two adjacent, , "they forma straight line., , on vertex.O and a common arm OC. They haven, adjacent angles. , angles form a linear pair, if their Non-common arms are two opposite rays ie,, , , , , , , , , , Observe the figure. Here ZAOB and ZBOC are two adjacent an;, , are two opposite rays i.e., they form.a straight line. Such a pair o}, , The angles in a linear pair are always supplementary., , Therefore the sum of the angles forming a linear pair is, , . Vertically Opposite Angles: : Bees :, , () Two angles formed by two intersecting lines havin,, vertically opposite angles., , (i) Observe the adjoining figure carefully. Here two lines 1, at the point O making four angles 21, 22, 23 and 44,, , S Whose non-common arms OA and OC, gles is called a linear pair., , , , , , , , 180°, , i NO common arm are said tobe ; ', , , , and m intersect each other, , 1 is vertically opposite t, 43 and 22 is vertically opposite to 24. ¥ opp 2, Gi) When two lines intersect, the vertically Opposite angles so formed are equal to, : each other,, , Scanned with CamScanner
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We want to prove that, 41= Z3and 22= 44.., , Now 21 + 24 = 180° ~f:, , , , , , , , Also 43 + 24 = 180° © oe ~ (i |, 23 and 24 form a linear pair) :, From (i) and (i), : :, 21+44=23+Z4, ALE ZR E., , Similarly, we can prove, , Scanned with CamScanner
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Transversal: A line which intersects [wo :, or more lines at different points is called a :, transversal to the given lines. :, In the figure, the transversal p intersect the lines :, Jand mm. i, , wf, In this figure, the A, line p intersects ~, se, , two lines | and OP, m, but it is not a oe, , a transversal, 9 *~ “Ame g, because line p does not intersect the two lines :, Tand m in two different points. :, , Angles Made by a Transversal: In fig. |, , , , a transversal p Pp, intersects two lines | oN j, and m at points A and i, , B_ respectively. The, , eight angles made by (Efe n, , this transversal with LI], the other two lines are, Zi, 22, 23, 24, 25, :, 26, £7 and 28, These angles can be classified «, as follows: :, 1) Interior angles: 23, 24, 45, 26, iy Exterior angles: 21, 22, 27, 28, (ity Corresponding angles: 71 and £5, 22, and 46, 43 and Z7 Z4and 28., , i “Alternate Interior Angles: 23 and 25,, , Z4and 26., fey Alternate Exterior Angles: 21 and 27,, Z2and 28. aye rubies ath yt, , Transversal of Parallel Lines: Two lines, , are parallel if they do not intersect at all., , The relation between the angles made by a, , transversal with two parallel lines given rise, , to quite intersecting results., , (i) ftwo parallel lines are cutby a transversal,, then each pair of corresponding angles, are equal,, , (ii) If two parallel lines are cut by a, transversal, then the angles of each pair, of alternate interior angles so formed are, equal,, , (iii) If two parallel lines are cut by a, transversal, then the sum of the interior, angles on the same side of the transversal, is 180°., , Checking for Parallel Lines:, , (i) When a transversal intersects two lines,, if the corresponding angles are equal, the, lines will be parallel. :, , Scanned with CamScanner
