Understanding Elementary Shapes

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.I Understanding Elementary Shapes Learn and Remember. A line segment is a portion of a line. If A and B are two points in a plane, the partab of the line through A and B is called a line segment AB.

1 .I Understanding Elementary Shapes Learn and Remember. A line segment is a portion of a line. If A and B are two points in a plane, the partab of the line through A and B is called a line segment AB. The points A and B are called the end-points of this segment. 2. A line segment has two end points and a line has no end point.. There is a unique line segment joining two given points A and B. 4. The line segment AB is the same as the line segment BA. 5. The length of a line segment AB is denoted by AB. 6. Two or more line segments having the same length are said to be congruent or equal. 7. m = 00 em yard = ft em = 0 mm ft = 2 inches km = 000 m mile = 760 yard. 8. The distance between two points is the same as the length of the line segment joining these points. 9. The part of a line that extends indefinitely in one direction from a given point 0, is called a ray. 0. A ray has only one end point and has no definite length. TEXTBOOK QUESTIONS SOLVED EXERCISE 5. Ql. What is the disadvantage in comparing line segments by mere observation? There may be chance of error due to improper viewing. Q2. Why is it better to use a divider than a ruler, while measuring the length of a line segment? The thickness of the ruler may cause difficulties in reading off' the length. However divider gives up accurate measurement. Q. Draw any line segment, say AB. Take any point C lying in between A and B. Measure the lengths of AB, BC and AC. Is AB=AC+CB?

50 SoL Q4. SoL MATHEMATICS-VI [Note: If A, B, C are any three points on a line, such thatac = AB, then we can be sure that C lies between A and B.] Yes.; ~ B AB = 6.5 em,ac = em, CB =.

2 50 SoL Q4. SoL MATHEMATICS-VI [Note: If A, B, C are any three points on a line, such thatac = AB, then we can be sure that C lies between A and B.] Yes.; ~ B AB = 6.5 em,ac = em, CB =.5 em AC + CB = em +.5 em = 6.5 em = AB. + CB If A, B, C are three points on a line such that AB = 5 em, BC = cm and AC = 8 em, which one of them lies between the other two? AC is the longest line segment, thus B is the point between A and C. Q5. Verify whether D is the mid point of AG. SoL AD = units, DG = units ABC 0 E F G (I I I I I I I I ) AD = DG. Thus D is mid-point Q6. IfB is the mid point of AC and C is the mid point of BD, where A, B, C, D lie on a straight line, say why AB = CD? SoL B is mid point of AC AB=BC C is mid point of BD BC=CD...() From () and (2) AB=CD. Q7. Draw five triangles and measure their sides. Check in each case, if the sum of the lengths of any two sides is always less than the third side. SoL Yes, sum of two sides of a triangle is always greater than the third side. EXERCISE 5.2 Ql. What fraction of a clockwise revolution does the hour hand of a clock turn through, when it goes from t09 4t07 7 to 0 (d) 2 to 9 (e) to 0 (f) 6t0 SoL "2 or two right angles. "4 or one right angle. "4 or one right angle. (e) "4 or three right angles....(2) (d) "4 or three right angles. (j) "4 or three right angles. UNDERSTANDING ELEMENTARY SHAPES Q2. Where will the hand of a clock stop if it Starts at 2 and ~akes 2 of a revolution, Qlockwise? Starts at 2 and makes 2 of a revolution, clockwise? Starts at 5 and makes "4 of a revolution, clockwise? (d) Starts at 5 and makes "4 of a revolution, clockwise?.' SoL At 6 At 8 At 8 (d) At 2. Q. Which direction will you face if you start facing East and make 2 of a revolui4»n clockwise? East and make 2 ofarevolution clockwise? West and make "4 ofarevolution ~~ey.\~' Nt't~. 5 r; ~.,, -, ~.' ;. -Soutb -& anti-clockwise? (d) South and make one full revolution? (Should we specify clockwise or anti-clockwise for this last question? Why not?). SoL West West North (d) South. (To answer (d), it is immaterial -whether we turn clockwise or antieloekwise, because one full revolution will bring us back to the original position.) Q4. What part of a revolution have you turned through if you stand facing East and turn clockwise to face north? South and turn clockwise to face east? "4 Q5. West and turn clockwise to face east? -, Find the number of right ang~es turned through by the hour hand of a clock when it goes from ' to 6 2 i to 8 (d) 0 to (e) 2 to 9 5 to (f) 2 to6

MATHEMATIcs-VI 52 --------- ------------------ One right angle. Two right angles. Two right angles. (d) One right angle. (e) Three right angles. (j) Two right angles. Q6.

3 MATHEMATIcs-VI One right angle. Two right angles. Two right angles. (d) One right angle. (e) Three right angles. (j) Two right angles. Q6. How many right angles do you make if you start facing South and turn clockwise to west? North and turn anti-clockwise to east? West and turn to west? (d) South and turn to north? One right angle. Three right angles. Four right angles. (d) Two right angles. Q7. Where will the hour hand of a clock stop if it starts? From 6 and turns through right angle? From 8 and turns through 2 right angles? From 0 and turns through right angles? (d) From 7 and turns through 2 straight angles? At 9 At 2 At 7 (d) At 7 EXERCISE 5. Ql. Match the following: (i) Straight angle (ii) Right angle (Hi) Acute angle (iv) Obtuse angle (v) Reflex angle Less than one-fourth a revolution!"ore than half a revolution Halfofarevolution (d) One-fourth a revolution (e) Between ~ and iofarevolution (f) One complete revolution. (0 -H, (io -t d, (iii) -t a, (iv) -t e, (v) -t. Q2. Classify each one of the following angles as right, straight, acute, obtuse, or reflex: UNDERSTANDING ELEMENTARY SHAPES 5 (d) (d) acute angle reflex angle EXERCISE 5.4 (e) obtuse angle (e) straight angle (j) right angle (f) acute angle. Ql. What is the measure of (i) a right angle? (ii) a straight angle? (i) A right angle is of 90. (ii) A straight angle is of 80. Q2. Say True or False: The measure of an acute angle < 90. The measure of an obtuse angle < 90. The measure of a reflex angle> ( Q > (d) The measure of one complete revolution = 60. (e) If mla = 5 and mlb = 5, titen mla > mlb. True False True (d) True Q. Write down the measures of Some acute angles.. Some obtuse angles. (give at least two examples of each.) 5, 20 0, 5. Q4. Measure the angles given below, using the protractor and write down the measure. (e) True. <,

54 MATHEMATIcs-VI (d) L SoL 40 0 90 (d) 60. Q5. Which angle has a large measure? First estimate and then measure MeasureofAngleA= Measure of Angle B = SoL LB has larger measure.

4 54 MATHEMATIcs-VI (d) L SoL (d) 60. Q5. Which angle has a large measure? First estimate and then measure MeasureofAngleA= Measure of Angle B = SoL LB has larger measure. Measure of angle A = 40 Measure of angle B = 65. ALB Q6. From these two angles which has larger measure? Estimate and then confirm by measuring them.,~. SoL Second angle has larger measure. Q7. Fill in the blanks with acute, obtuse, right or straight: An angle whose measure is less than that of a right angle is _ An angle whose measure is greater than that of a right angle is _ An angle whose measure is the sum of the measures of two right angles is. - (d) When the sum of the measures of two angles is that of a right angle, then each one of them is --- (e) When the sum of the measures of two angles is that of a straight angle and if one of them is acute then the other should be _ An angle whose measure is less than that of a right angle is acute angle. An angle whose measure is greater than that of a right angle is obtuse angle. UNDERSTANDING ELEMENTARY SHAPES 55 An angle whose measure is the sum ofthe measures oftwo right angles is straight angle. (d) When the sum of the measures of two angles is that of a right angle, then each one of them is acute angle. (e) When the sum of the measures oftwo angles is that ofa straight angle and if one ofthem is acute then the other should be obtuse angle. Q8. Find the measure of the angle shown in each figure. (First estimate with your eyes and then find the actual measure with a protractor). (i) 0 (ii) 20 (iii) 60 (iv) 50. Q9. Find the angle measure between the hands of the clock in each figure a.m. (i) 90 (right angle) QI0. Investigate:.00 p.m p.m. In the given figure, the angle measure 0. Look at the same figure through a magnifying glass. Does the angle becomes larger? Does the size of the angle change? No, the size of angle will be same. (ii) 0 (acute angle) (iii) 80 (straight angle).

56 ~~THEMATICS-Vl UNDERSTANDING ELEMENTARY SHAPES ---------,------_...- 57 QU. Measure and classify each angle: B Q4. Study the diagram.

5 56 ~~THEMATICS-Vl UNDERSTANDING ELEMENTARY SHAPES ,------_ QU. Measure and classify each angle: B Q4. Study the diagram. The line I is perpendicular to line m, A ABC D E F G H ~ I I I I I I I I I ) o Angle LAOB LAOC LBOC LDOC LDOA LDOB Measure Type Angle LAOB LAOC LBOC LDOC LDOA LDOB Measure Type Acute Obtuse Right Right Obtuse Straight Is CE = EG? Does PE bisect CG? Identify any two line segments for which PE is the perpendicular bisector. (d) Are these true? (i) AC > FG (ii) CD = GH (iii) BC < EH. Yes, both measure 2 units. Yes, because CE = EG. DF and cd, BH (d) (i) True (ii) True (iii) True. EXERCISE 5.6 p m EXERCISE 5.5 Q. Which of the following are models for perpendicular lines: The adjacent edges ofa table top. The lines of a railway track. The line segments forming the letter 'L'. (d) The letter V. Perpendicular. Perpendicular. Not perpendicular. (d) Not perpendicular. Q2. Let PQ be the perpendicular to the line segment XY. Let PQ and XY intersect in the point A. What is the measure of LPAY? x LPAY= 90. Q. There are two "set-squares" in your box. What are the measures of the angles that are formed at their corners? Do they have any angle measure that is common? One has 45 _90-45 angles and other has LBoo-L900-L0. They have 90 as common angle. p A Q y Ql. Name the types of following triangles: Triangle with lengths of sides 7 em, 8 em and 9 cm. MBC withab = 8.7 cm,ac = 7 cm and BC = 6 cm. APQR such that PQ = QR = PR = 5 em, (d) LIDEFwith mld = 90. (e) AXYZwith mly = 90 and XY = YZ. (f) M.MN with mll = 0, mlm = 70 and mln = 80. Scalene triangle. Scalene triangle. Equilateral triangle. (d) Right-angled triangle. (e) Isosceles right-angled triangle. (f) Acute-angled triangle. Q2. Match the following:. Measures of Triangle Types of Triangle (i) sides of equal length Scalene (ii) 2 sides of equal length Isosceles right angled (iii) All sides are of different length Obtuse angled (iv) acute angles (d) Right angled (v) right angle (e) Equilateral (vi) obtuse angle (f) Acute-angled (vii) right angle with two sides (g) Isosceles. of equal length (i) ~ (e), (ii) ~ (g), (iii) ~, (iv) ~ (f), (v) ~ (d), (vi) ~, (vii) ~.

6 58 MATHEMATlCS-VI Q. Name each of the following triangles in two different ways: 0em Q4. (You may judge the nature ofthe angle by observation) 5cm 7em 8em 5 em 7em 0em (d) (e) (f) Acute-angled and isosceles {b) Right-angled and scalene Obtuse-angled and isosceles (d) Right-angled and isosceles (e) Equilateral and acute angled (f) Obtuse-angled and scalene. Try to construct triangles using match sticks. Some are shown here. Can you make a triangle. matchsticks? 4 matchsticks? 5 matchsticks? (d) 6 matchsticks? (Remember you have to use all the available matchsticks in each case) Name the type oftriangle in each case. If you cannot make a triangle, think of reasons for it. with 0em \ i \-7 \/ V 6 /\\ UNDERSTANDING ELEMENTARY SHAPES 59 Matchsticks /\ This is an acute triangle and it is possible with matchsticks to make a triangle because sum of two sides is greater than third side. 4 Matchsticks Ị This is a square, hence with four matchsticks we cannot make triangle. 5 Matchsticks \ U L\7 This is an acute triangle and it is possible to make triangle with five matchsticks, in this case sum of two sides is greater than third side. (d) 6 Matchsticks /\ L~ This is an acute triangle and it is possible to make a triangle with help of 6 matchsticks because sum of two sides is greater than third side. EXERCISE 5.7 Ql. Say True or False: Each angle of a rectangle is a right angle. The opposite sides ofa rectangle are equal in length. The diagonals of a square are perpendicular to one another. (d) All the sides of a rhombus are of equal length. (e) All the sides of a parallelogram are of equal length. (n The opposite sides of a trapezium are parallel. True True True (d) True (e) False (f) False.

MATHEMATlCS-VI 60 ----~-.------------ -- Q2. Give reasons for the following: A square can be thought of as a special rectangle. A rectangle can be thought of as a special parallelogram.

7 MATHEMATlCS-VI ~ Q2. Give reasons for the following: A square can be thought of as a special rectangle. A rectangle can be thought of as a special parallelogram. A square can be thought of as a special rhombus. (d) Squares, rectangles, parallelograms are all quadrilaterals. (e) Square is also a parallelogram. Because its all angles are right angle and opposite sides are equal. Because its opposite sides are equal and parallel. (e) Because its four sides are equal and diagonals are perpendicular to each other. (d) Because all of them have four sides. (e) Because its opposite sides are equal and parallel. Q. A figure is said to be regular if its sides are equal in length and angles are equal in measure. Can you identify the regular quadrilateral? A square is a regular quadrilateral. EXERCISE 5.8 Q. Examine whether the following are polygons. If anyone among them is not, say why? 06 ~) ~) ~ ~ It is not a closed figure. Only 'b' is a polygon. Not made from line segment. (d) Not made from only line segment, having curved surface. Q2. Name each polygon: ~CJO UNDERSTANDING ELEMENTARY SHAPES 6 Quadrilateral Triangle Pentagon (d) Octagon. Q.. Draw a rough sketch of a regular hexagon. Connecting any three of its vertices, draw a triangle. Identify the type of the triangle you have drawn. F A B, E 0 ABCDEF is a regular hexagon and l thus formed by joining AEF is isosceles triangle. Q4. Draw a rough sketch of a regular octagon. (Use squared paper if you wish) Draw a rectangle by joining exactly four of the vertices of the octagon. Q5. ABCDEFGH A C Hf C G\ ;0 F E is a regular octagon and CDGH is a rectangle. A diagonal is a line segment that joins any two vertices of the polygon and is not a side of the polygon. Draw a rough sketch of a pentagon and draw its diagonals. A, \, \ -, \ '>,... \ "..." I.,...>.: 'l... e : \ I' E" \ I "~ \ I...-C \ I \ I \ I ~ D ABCDE is the required pentagon and its diagonals are AD, AC, BE, BD. (d)

8 62 MATHEMATlcs-VI EXERCISE 5.9 Q.. Match the following: Cone (i) ~ Sphere (ii) ~ Cylinder (iii) ~~I (d) Cuboid (iv)...::' -4 (ii) -4 (iv) -4(v) (d) -4 (iii) (e)-4 (i) (e) Pyramid (v) Give two new examples of each shape. Example: Cone -4 Ice cream, birthday cap. Sphere -4 Cylinder -4 Cuboid -4 Ball, Laddu Can, powder box Match box, pencil box Pyramid -4 Prism, dome of temple. Q2. What shape is Your instrument box? Abrick? A match box? (d) Aroad-roller? (e) A sweet laddu? Cuboid. Cuboid. (e) Sphere. Cuboid. (d) Cylinder. 00

Understanding Elementary Shapes

Understanding Elementary Shapes Chapter 5 5.1 Introduction All the shapes we see around us are formed using curves or lines. We can see corners, edges, planes, open curves and closed curves in our surroundings.