Lines and Angles BASIC GEOMETRICAL CONCEPTS (AXIOMS, THEOREMS AND COROLLARIES)

Transcription

1 4 CHAPTER We are Starting from a Point but want to Make it a Circle of Infinite Radius BASIC GEOMETRICAL CONCEPTS (AXIOMS, THEOREMS AND COROLLARIES) Axioms Examples STATEMENTS Examples The basic facts which are taken for granted, without proof, are called axioms (i) Halves of equal are equal (ii) The whole is greater than each of its parts (iii) A line contains infinitely many points A sentence which can be judged to be true or false is called a statement (i) The sum of the angles of a triangle is 18, is a true statement (ii) The sum of the angles of a quadrilateral is 18, is a false statement (iii) x + 1 > 15 is a sentence but not a statement Theorems A statement that requires a proof, is called a theorem Establishing the truth of a theorem is known as proving the theorem Examples (i) The sum of all the angles around a point is 36 (ii) The sum of the angles of a triangle is 18 CAROLLARY : A statement, whose truth can easily be deduced from a theorem, is called its corollary EUCLID S FIVE POSTULATES 1 A straight line may be drawn from any point to any other point 2 A terminated line can be produced indefinitely 3 A circle can be drawn with any center and any radius 4 All right angle are equal to one another 5 If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the angles taken together are less than two right angles Later on the fifth postulate was modified as under For every line L and for every point P not lying on L, there exists a unique line M, passing through P and parallel to L Clearly, two distinct intersecting lines cannot be parallel to the same line SOME TERMS RELATED TO GEOMETRY POINT A point is an exact location A fine dot represents a point We denote a point by a capital letter A, B, P, Q, etc In the given figure, P is a point Line segment The straight path between two points A and B is called the line segment AB The points A and B are called the end points of the line segment AB A line segment has a definite length /

2 The distance between two points A and B is equal to the length of the line segment AB RAY A line segment AB when extended indefinitely in one direction is the ray AB Ray AB has one end point A A ray has no definite length A ray cannot be drawn, it can simply be represented on the plane of a paper To draw a ray would mean to represent it LINE A line segment AB when extended indefinitely in both the directions is called the line AB A line has no end points A line has no definite length A line cannot be drawn, it can simply be represented on the plane of a paper To draw a line would mean to represent it Sometimes, we lable lines by small letters l, m, n, etc INCIDENCE AXIOMS ON LINES (i) A line contains infinitely many points (ii) Through a given point, infinitely many liens can be drawn (iii) One and only one line can be drawn to pass through two given points A and B COLLINEAR POINTS Three or more than three points are said to be collinear, if there is a line which contains them all In the given figure A,B,C are collinear points, while P,Q,R are non-collinear INTERSECTING LINES Two lines having a common point are called intersecting lines In the given figure, the lines AB and C intersect at a point O CONCURRENT LINES Three or more lines intersecting at the same point are said to be concurrent In the given figure, lines l, m, n pass through the same point P and therefore, they are concurrent PLANE A plane is a surface such that every point of the line joining any two points on it, lies on it Examples The surface of a smooth wall; the surface of the top of the table; the surface of a smooth blackboard; the surface of a sheet of paper etc, are close examples of a plane These surfaces are limited in extent but the geometrical plane extends endlessly in all directions Parallel Lines Two lines l and m in a plane are said to be parallel, if they have no point in common and we write, l m The distance between two parallel lines always remains the same /

3 Questions 1 (i) How many lines can be drawn to pass through a given point? (ii) How many lines can be drawn to pass through two given points? (iii) In how many points can the two lines at the most intersect? (iv) If A, B, C are three collinear points, name all the line segments determined by them 2 Which of the following statements are true? (i) A line segment has no definite length (ii) A ray has no end point (iii) A line has a definite length (iv) A line AB is the same as line BA (v) (vi) A ray AB is the same as ray BA Two distinct points always determine a unique line (vii) Three lines are concurrent if they have a common point (viii) Two distinct lines cannot have more than one point in common (ix) (x) (xi) Two intersecting liens cannot be both parallel to the same line Open half-line OA is the same thing as ray Two lines may intersect in two points OA (xii) Two lines l and m are parallel only when they have no point in common ANGLES AND THEIR PROPERTIES ANGLE Two rays OA and OB having a common end point O form angle AOB, written as AOB OA and OB are called the arms of the angle and O is called its vertex INTERIOR OF AN ANGLE The interior of AOB is the set of all points in its plane, which lie on the same side of OA as B and also on the same side of OB as A, eg, P is a point in the interior of AOB Any point on any arm or vertex is said to lie on the angle, eg, Q is a point on AOB EXTERIOR OF AN ANGLE The exterior of an angle AOB is the set of all those points in its plane, which do not lie on the angle or in its interior In the given figure, R is a point in the exterior of AOB /

4 MEASURE OF AN ANGLE The amount of turning from OA to OB is called the measure of AOB, written as m AOB An angle is measured in degrees denoted by AN ANGLE OF 36 If a ray OA starting from its original OA, rotates about O, in the anticlockwise direction and after making a complete revolution it comes back to its original position, we say that it has rotated through 36 degrees, written as 36 This complete rotation is divided into 36 equal parts Each part measures 1 1 = 6 minutes, written as 6 1 = 6 seconds, written as 6 We use a protractor to measure an angle KINDS OF ANGLE (i) (ii) (iii) (iv) (v) (vi) RIGHT ANGLE An angle whose measure is 9 is called a right angle ACUTE ANGLE An angle whose measure is more than but less than 9 is called an acute angle OBTUSE ANGLE An angle whose measure is more than 9 but less than 18 is called an obtuse angle STRAIGHT ANGLE An angle whose measure is 18 is called a straight angle REFLEX ANGLE An angle whose measure is more than 18 but less than 36 is called a reflex angle COMPLETE ANGLE An angle whose measure is 36 is called a complete angle EQUAL ANGLES Two angles are said to be equal, if they have the same measure Bisector of an angle A ray OC is called the bisector of AOB, if m AOC = m BOC 1 In this case, AOC = BOC = AOB 2 COMPLEMENTARY ANGLES Two angles are said to be complementary, if the sum of their measures is 9 Two complementary angles are called the complement of each other Example Angles measuring 55 and 35 are complementary angles /

5 SUPPLEMENTARY ANGLES Two angles are said to be supplementary, if the sum of their measures is 18 Example Angles measuring 62 and 118 are supplementary angles Example 1 Find the measure of an angle which is 24 more than its complement Solution Let the measure of the required angle be x Then measure of its complement = (9 x) x (9 x) = 24 2x = 114 x = 57 Hence, the measure of the required angle is 57 Example 2 Find the measure of an angle which is 32 less than its supplement Solution Let the measure of the required angle be x Then measure of its complement = (18 x) x (18 x) = 32 2x = 148 x = 74 Hence, the measure of the required angle is 74 Example 3 Find the measure of an angle, if six times its complement is 12 supplement Solution Let the measure of the required angle be x Then measure of its complement = (9 x) Measure of its supplement = (18 x) 6 (9 x) = 2 (18 x) x = 36 2x 12 4x = 192 x = 48 less than twice its /

6 IX ACADEMIC QUESTIONS Subjective Assignment 1 1 Define the following terms: (i) Angle (iii) Obtuse angle (v) Complementary angles (ii) Interior of an angle (iv) Reflex angle (vi) Supplementary angles 2 Find the complement of each of the following angles (i) 58 (ii) 16 1 (iii) of a right angle 2 3 Find the supplement of each of the following angles (i) 63 (ii) (iii) of a right angle 5 4 Find the measure of an angle which is 36 more than its complement 5 Find the measure of an angle which is 25 less than its supplement 6 Two supplementary angles are in the ratio 3 : 2 Find the angles 7 Find the measure of an angle, if seven times its complement is 1 less than three times its supplement 8 In Fig lines PQ and RS intersect each other at point O If POR : ROQ = 5 : 7, find all the angles P R O S Q 9 In Fig ray OS stands on a line POQ Ray OR and ray OT are angle bisectors of POS and SOQ, respectively If POS = x, find ROT R S T P O Q 1 In Fig OP, OQ, OR and OS are four rays Prove that POQ + QOR + SOR + POS = /

7 11 11 In Fig lines AB and CD intersect at O If AOC + BOE = 7 and BOD = 4, find BOE and reflex COE In Fig lines XY and MN intersect at O If POY = 9 and a : b = 2 : 3, find c 13 In Fig PQR = PRQ, then prove that PQS = PRT 14 In Fig if x + y = w + z, then prove that AOB is a line 15 In Fig POQ is a line Ray OR is perpendicular to line PQ OS is another ray lying between rays OP and OR Prove that ROS = 2 1 ( QOS POS) 16 It is given that XYZ = 64 and XY is produced to point P Draw a figure from the given information If ray YQ bisects ZYP, find XYQ and reflex QYP SOME ANGLE RELATIONS ADJACENT ANGLES Two angles are called adjacent angles, if (i) they have the same vertex, (ii) they have a common arm and (iii) their non-common arms are on either side of the common arm In the given figure, AOC and BOC are adjacent angles having the same vertex O, a common arm OC and their noncommon arms OA and OB on either side of OC LINEAR PAIR OF ANGLES Two adjacent angles are said to form a linear pair of angles, if their non-common arms are two opposite rays /

8 In the adjoining figure, AOC and BOC are two adjacent angles whose non-common arms OA and Ob are two opposite rays, ie, BOA is a line AOC and BOC form a linear pair of angles SOME RESULTS ON ANGLES RELATIONS Theorem 1 If a ray stands on a line then the sum of the adjacent angles so formed is 18 Given A ray CD stands on a line AB such that ACD and BCD are formed To prove ACD BCD 18 Construction Draw CE AB Proof ACD ACE ECD (i) and BCD BCE ECD (ii) Adding (i) and (ii), we get : ACD BCD = ( ACE ECD ) ( BCE ECD ) = ACE BCE = (9 + 9 ) = 18 [ ACE BCE 9 ] Hence, ACD BCD 18 REMARK we may state the above theorem as the sum of the angles of a linear pair is 18 COROLLARY 1 Prove that the sum of all the angles formed on the same side of a line at a given point on the line is 18 Given AOB is a straight line and rays OC, OD and OE stand on it, forming AOC, COD, DOE and EOB To prove AOC COD DOE EOB 18 Proof Ray OC stands on line AB AOC COB = 18 ( AOC ( COD DOE EOB) 18 [ COB COD DOE EOB ] AOC COD DOE EOB 18 Hence, the sum of all the angles formed on the same side of line AB at a point O on it is 18 COROLLARY 2 Prove that the sum of all the angles around a point is 36 Given A point O and the rays OA, OB, OC, OD and OE make angles around O To Prove AOB BOC COD DOE EOA 36 Construction Draw a ray OF opposite to ray OA PROOF Since ray OB stands on line FA, we have : AOB BOF = 18 [linear pair] AOB BOC COF = 18 (i) [ BOF BOC COF ] Again, ray OD stands on line FA FOD DOA = 18 [linear pair] or FOD DOE EOA = 18 [ DOA DOE EOA ] Adding (i) and (ii), we get : AOB BOC COF FOD DOE EOA = 36 AOB BOC COD DOE EOA /

9 [ COF FOD COD ] Hence, the sum of all the angles, around a point O is 36 VERTICALLY OPPOSITE ANGLES Two angles are called a pair of vertically opposite angles, if their arms form two pairs of opposite rays Let two lines AB and CD intersect at a point O Then, two pairs of vertically opposite angle are formed : (i) AOC and BOD (ii) AOD and BOC Theorem 2 If two lines intersect then the vertically opposite angle are equal Given Two lines AB and CD intersect at a point O To Prove (i) AOC BOD, (ii) AOD BOC PROOF Since ray OA stands on line CD, we have AOC BOD 18 [linear pair] Again ray OD stands on line AB AOD BOD 18 [linear pair] AOC AOD AOD BOD [each equal to 18 ] AOC BOD Similarly, AOD BOC Example 4 In the adjoining figure, AOB is a straight line Find AOC and BOD Solution Since AOB is a straight line, the sum of all the angles on the same side of AOB at a point O on it, is 18 x (2x 2) = 18 3x = 135 x = 45 AOC 45 and BOD = (2 45 2) = 7 Example 5 In the adjoining figure, what value of x will make AOB a straight line? Solution AOB will be a straight line, if A OC BOC 18 (3x + 5) + (2x 25) = 18 5x = 2 x = 4 Hence, x = 4 will make AOB a straight line THE ANGLES FORMED WHEN A TRANSVERSAL CUTS TWO LINES Let AB and CD be two lines, cut by a transversal t Then, the following angles are formed (i) Pairs of corresponding angles : ( 1, 5 ); ( 4, 8 ); ( 2, 6 ) and ( 3, 7 ) (ii) Pairs of alternate interior angles : ( 3, 5 ) and ( 4, 6 ) (iii) Pairs of consecutive interior angles (allied angles or conjoined angles) : ( 4, 5 ) and ( 3, 6 ) REMARKS We shall abbreviate as follows : (i) Corresponding Angles as corres s (ii) Alternate Interior Angles as Alt Int s /

10 (iii) Consecutive Interior Angles as Co Int s CORRESPONDING ANGLES AXIOM If a transversal cuts two parallel lines then each pair of corresponding angles are equal Conversely, if a transversal cuts two lines, making a pair of corresponding angles equal, then the lines are parallel Thus, whenever AB CD are cut by a transversal t, then 1 5 ; 48 ; 2 6 and 3 7 On the other hand, if a transversal t cuts two lines AB and CD such that ( 15 ) or ( 48 ) or ( 26 ) or ( 37 ) then AB CD Theorem 3 If a transversal intersects two parallel lines then alternate angles of each pair of interior angles are equal Given AB CD and a transversal t cuts AB at E and CD at F, forming two pairs of alternate interior angles, namely ( 3, 5 ) and ( 4, 6 ) To Prove 3 5 and 4 6 PROOF We have 3 1 (vert opp s ) and 1 5 (corres s ) 3 5 Again, 4 2 (vert opp s ) and 2 6 (corres s ) 4 6 Hence, 3 5 and 4 6 Theorem 4 If a transversal intersects two parallel lines then each pair of consecutive interior angles are supplementary Given AB CD and a transversal t cuts AB at E and CD at F, forming two pairs of consecutive interior angles, namely ( 3 6 ) and ( 4 5 ) TO PROVE 3 6 = 18 and PROOF Since ray EF stands on line AB, we have (linear pair) But, 4 6 (Alt Int s ) Again, since ray FE stands on line CD, We have But, 6 4 (Alt Int s ) /

11 Hence, and Theorem 5 (Converse of Theorem 1) if a transversal intersects two lines, making a pair of alternate interior angles equal, then the two lines are parallel Given A transversal t cuts two lines AB and CD at E and F respectively such To Prove AB CD PROOF We have 35 (given) But, 31 (vert opp s ) 1 5 But, these are corresponding angles AB CD (by corres s axiom) Theorem 6 (Converse of Theorem 2) If a transversal intersect two lines in such a way that a pair of consecutive interior angles are supplementary then the two lines are parallel Given A transversal cuts two lines AB and CD at E and F respectively such that To Prove AB CD PROOF Since ray EB stands on line t, we have (linear pair) and (given) This gives, 15 But, these are corresponding angles AB CD (by corres s axiom) /

12 IX ACADEMIC QUESTIONS Subjective Assignment 2 1 Prove that the bisectors of the angles of a linear pair are at right angles 2 In the adjoining figure, AOB is a straight line Find the value of x 3 In the adjoining figure, AOB is a straight line Find the value of x Hence, find AOC, COD and BOD 4 In the adjoining figure, x : y : z = 5 : 4 : 6 If XOY is a straight line, find the values of x, y and z 5 If the bisectors of a pair of corresponding angles formed by a transversal with two given lines are parallel, prove that the given lines are parallel 6 In the given figure, AB CD Find the value of x 7 In the given figure, AB CD Find the value of x 8 For what value of x will the lines l and m be parallel to each other? /

13 9 In Fig if PQ RS, MXQ = 135 and MYR = 4, find XMY 1 If a transversal intersects two lines such that the bisectors of a pair of corresponding angles are parallel, then prove that the two lines are parallel 11 In Fig AB CD and CD EF Also EA AB If BEF = 55, find the values of x, y and z 12 In Fig if AB CD, CD EF and y : z = 3 : 7, find x 13 In Fig if AB CD, EF CD and GED = 126, find AGE, GEF and FGE 14 In Fig if PQ ST, PQR = 11 and RST = 13, find QRS [Hint : Draw a line parallel to ST through point R] 15 In Fig if AB CD, APQ = 5 and PRD = 127, find x and y 16 In Fig if QT PR, TQR = 4 and SPR = 3, find x and y /

14 17 In Fig sides QP and RQ of PQR are produced to points S and T respectively If SPR = 135 and PQT = 11, find PRQ 18 In Fig X = 62, XYZ = 54 If YO and ZO are the bisectors of XYZ and XZY respectively of XYZ, find OZY and YOZ 19 In Fig if AB DE, BAC = 35 and CDE = 53, find DCE 2 In Fig 642, if lines PQ and RS intersect at point T, such that PRT = 4, RPT = 95 and TSQ = 75, find SQT 21 In Fig 643, if PQ PS, PQ SR, SQR = 28 and QRT = 65, then find the values of x and y 22 In Fig 644, the side QR of PQR is produced to a point S If the bisectors of PQR and PRS meet at point T, then prove that QTR = 2 1 QPR /

15 XI SCIENCE & DIP ENTRANCE Multiple Choice Question Assignment 3 1 In the figure the lines AB and BD lie in a straight line If ABC 3x 8 and DBC x 4, then what is the value of x? 44 (b) 52 (c) 6 (d) 2 In the above figure if 3 DBC 2ABC, then what is value of DBC? 36 (b) 48 (c) 72 (d) 81 3 In the figure if POR and QOR form a linear pair If a b 8, then angle 'a' is 8 (b) 13 (c) 14 (d) 4 In the figure AB is a straight line If AOC x 2, COD 2x 15 and BOD x 1 45 (b) 64 15, then COD is 18 (c) 75 (d) 9 5 In the above figure if AOC : COD: BOD 3: 4: 2, then AOC is 4 (b) 6 In the following figure y x 12, then z is equal to 75 (b) 5 (c) 6 (d) 7 AOB 9 and CD is a straight line If 9 (c) 1 (d) 12 7 In the above Fig if 2 (b) z 15 what is the value of 4 (c) y? 6 (d) 8 In the figure BOE 9 AB and CD are straight lines If x : y 2 : 3, then the value of z is 76 (b) 6 (c) 118 (d) 9 In the above figure if z 3x, then y is equal to 45 (b) 6 (c) 75 (d) 1 In the figure AB and CD are straight lines If x m and the value of z is 45 (b) 12 (c) 135 (d) n 9, then /

16 11 In the above figure if is 4 (b) 12 In the figure, the value of x is y 4 and n 2m 1 5 (c), then the value of 6 (d) 3 o (b) 4 o (c) 45 o (d) 7 5 x 13 In the figure, AOB is a straight line OP and OQ are bisectors of BOC and AOC respectively, then the value of POQ is 7 (b) 32 (c) 9 o (d) What is the value of x in the figure? 6 o (b) 45 o (c) 3 o (d) 15 o 15 What is the complement angle of 79 o (b) 16 What is the supplement angle of 123 o (b) 82? 5 (c) 123? 6 (c) 17 If angles 2a 1 and 11 is equal to 25 (b) 18 An angle is 52 o 3 (d) 57 (d) 8 45 a are complementary angles then a 37 (c) 45 (d) more than its complement What is its measure? (b) 55 (c) 62 (d) 19 The measure of an angle is twice the measure of its supplementary angle What is the value of greatest angle? 6 (b) 9 (c) 12 (d) When two supplementary angles differ by smaller angle? 4 (b) 45 (c) 32, what is the value of 6 (d) 21 How many degrees are there in an angle which equals one-fifth of its complement? 18 o (b) 15 (c) 12 (d) 1 22 How many degrees are there in an angle which equals two-third of its supplement? 72 (b) 9 (c) 12 (d) How many degrees are there in an angle whose complement is onefourth of its supplement? /

17 3 (b) 6 (c) 45 (d) 9 In the figure PQ is an incident ray and QR the reflected ray If PQR 124, then RQB is 25 (b) 28 (c) 32 (d) 24 The measure of an angle whose supplement is three-times as large as its complement is 3 (b) 45 (c) 6 (d) If two angles are complementary of each other, then each angle is An obtuse angle (b) A right angle (c) An acute angle (d) A supplementary angle 26 In the figure if AOB is a straight line, then the value of x is 9 o (b) 45 (c) 1 22 (d) 2 27 An angle is greater than 18 but less than 36 is called An acute angle (b) An obtuse angle (c) An adjacent angle (d) A reflex angle 28 If AB CD and 2a 5b 1 then b is equal to 5 (b) 6 (c) 75 (d) In the given figure, where AB CD EF if x 2z 15 the value of y? 6 (b) 9 (c) 12 (d), then what is 13 3 From the figure, where AB CD, what is the value of x? 18 (b) 21 (c) 24 (d) From the figure calculate the value of x 3 o (b) 45 o (c) 6 o (d) 75 o 32 If AB CD as shown in the figure, calculate the value of x (c) 5 (b) 6 25 (d) 33 In the adjoining figure BCD is equal to 32 ABC 1, 4 (b) 6 EDC 12 and AB DE, then /

18 (c) 8 (d) 1 34 In the given figure If AB CD, then FXE is equal to 3 (b) 5 (c) 7 (d) 8 35 If two parallel lines are intersected by a transversal line then the bisectors of the interior angles form a Rhombus (b) Parallelogram (c) Square (d) Rectangle 36 In the figure AB CD value of DCE? BAE 1 and 1 (b) 125 (c) 75 (d) In the figure m n and p q If 15 (b) 9 (c) 75 (d) AEC 25, what is the 1 75, then 2 is equal to 6 38 Two parallel lines AB and CD are intersecting by a transversal line EF at M and N respectively The lines MP and NP are the bisectors of interiors angles BMN and DMN on the same side of transversal line, then MPN is equal to 45 (b) 6 (c) 9 (d) In the figure arms BA and BC of ABC are respectively parallel to arms ED and EF of DEF, then ABC DEF is 9 (b) 12 (c) 16 (d) /

19 ANSWER Assignment 1 2 (i) 32 (ii) 74 (iii) 45 3 (i) 117 (ii) 42 (iii) o 6 18 o, 72 o 7 25 o 8 POR = QOS = 75, ROQ = POS = o 11 BOE = 3 o and reflex COE = 25 o 12 c = 126 o 16 QYP = 32 o Assignment o 3 x = 32 o, AOC = 13 o, COD = 45 o, BOD = 32 o 4 x = 6 o, y = 48 o, z = 72 o 6 x = 4 o 7 x = 5 o 8 (i) x = 3 o (ii) x = 5 o 9 XMY = 85 o 11 x = 13 o, y = 13 o, z = 4 o 12 x = 126 o 13 AGE = 126 o, GEF = 36 o, FGE = 54 o 14 QRS = 6 o 15 x = 5 o, y = 77 o 16 x = 5 o, y = 8 o o o, 121 o o 2 6 o o, 53 o Assignment 3 1a 2c 3b 4b 5c 6d 7b 8d 9a 1c 11b 12b 13c 14d 15d 16c 17b 18a 19c 2d 21b 22a 23b 24b 25b 26c 27b 28d 29a 3d 31d 32a 33c 34a 35d 36d 37b 38a 39c 4d /