Allied Publication Pvt. Ltd. Sitapaila, Kathmandu Ph. No.: 01-5388827 The Leading MATHS 5 Author Ashok Dangol M.Ed., Maths (TU) ALLIED Prepared on New Curriculum Issued by CDC, Sanothimi, Bhaktapur, Nepal
Allied The Leading MATHS 5 Publisher Allied Publication Pvt. Ltd. Sitapaila, Kathmandu Phone : 01-5378629, 5388827 Written by Ashok Dangol M.Ed., Maths (TU) Special Thanks Nabaraj Pathak Lalbabu Prasad Yadav Jit Bahadur Khanal Govinda Prasad Pokharel Copyright All rights reserved with the copyright holder. Edition First - 2080 (5000 pcs.) Computer Icon Design House #9849098999 Printed in Nepal
iv Creative Maths -VI Approved by CDC, Nepal PREFACE This Allied The Leading Mathematics - 5 is basically meant for making the teachers and taught active while teaching and learning mathematics. The contents and extent of the series are strictly contained and arranged in accordance with the new vision and mission of the latest New Curriculum 4-5 of Basic Level of CDC, Nepal. This series is basically an outcome of my untiring effort and patience. The long and dedicated service in teaching and popularization of mathematics has been a great asset in preparing this series. It has been designed as a textbook for English medium private and government school students with a new approach. This book provides maximum benefit to both teachers and students because of the following unique features: Unique Features of this Book Ö Arranged especially focusing on child psychology of teaching and learning mathematics which are based on the Areas of Basic Level Crriculum 6-8. Ö Prepared with the firm belief that “Mathematics begins at Home, grows in the Surroundings and takes shape in School (HSS Way)”, and sincere attempts have been made to make the learner, the teacher and reader feel “Mathematics is Fun, Mathematics is Easy and Mathematics is Everywhere (FEE Concept)”. Ö Written the focus of students’ activities and easily perform teaching-learning activities for teachers. Ö Well arranged the four colours of the whole book supports to find easily Units, Chapters, Lessons, Examples, Practices, and other topics from Content. Ö Every Theme begins with its estimated teaching hours (Theory + Practical), competency, learning outcomes, Warm-Up for pre-knowledge. Ö Highlighted the important terms, notes and key points. Ö Included sufficiently all types of Classwork Examples and Home Assessments from simple to complex with suitable figures and reasons. Ö Included Project Works as self-practice to the students at home for some days' activities to memories long time about the entire chapters. Ö Included Mixed Practice and Confidence Level Tests as self-evaluations for Success and Competent by themselves. Ö Available Individual Practical Evaluation Sheet at the end.
PREFACE It is very much hoped that with all the above features, this book will be found really fruitful by teachers and students alike. Thank Allied Publication Pvt. Ltd., Kathmandu, Nepal for taking responsibility for publishing this book, Nariswor Gautam for language editing, Dev Krishna Maharjan for an attractive art of pictures, and Binod Bhandari for an attractive design. I would like to extend my sincere gratitude to the persons whose ideas or creations are directly or indirectly incorporated into the text. I would like to extend my thanks to the teachers and the students who helped me to verify the answers and to check the manuscript of this book. Also, many thanks to the schools that applied this book and suggested it to me. Finally, I heartily welcome criticisms, feedbacks and suggestions from readers so that it may appear with revise from in the coming edition and will be gratefully and thankfully acknowledged and honored.
CONTENTS UNIT - I: GEOMETRY 7 Chapter 1: Lines and Angles 8 Chapter 2: Solid Objects 59 UNIT - II: ARITHMETIC 69 Chapter 3: Numbers Sense 70 Chapter 4: Simplification 98 Chapter 5: Fractions 105 Chapter 6: Decimal 134 Chapter 7: Percentage 149 UNIT - III:MEASUREMENT 164 Chapter 8: Time 165 Chapter 9: Distance 179 Chapter 10: Capacity 192 Chapter 11: Weight 205 Chapter 12: Perimeter, Area And Volume 219 NIT - IV: STATISTICS 247 Chapter 13: Bill And Budget 248 Chapter 14: Bar Diagram 263 UNIT - IV:ALGEBRA 272 Chapter 15: Algebraic Expressions 273 Chapter 16: Equation Of One Variable 289
Geometry 7 GEOMETRY UNIT I Estimated Teaching Periods : 19 Marks Weightage : 9 COMPETENCY Measurement and construction of lines and angles, and identification of different parts of solid objects CHAPTERS 1. Lines and Angles 2. Solid Objects LEARNING OUTCOMES After completion of this content area, the learner is expected to be able to: measure and draw the angles from 0° to 180° by using protractor. measure the interior angles of the given triangle and quadrilateral. classify the right angle, acute angle and obtuse angle. draw perpendicular line and parallel line in square grid paper. count the number of vertices, edges and surfaces of the cube and cuboid.
8 The Leading Mathematics - 5 What is geometry? Discuss. What is point in geometry? What is line ? What are the types of lines? Can you measure the line and line segment ? How are the edges of room, the board, book and duster? How are the corners of the board, book and duster ? What are parallel, intersecting and perpendicular lines ? What is an angle ? Can you measure the angles ? Can you draw the given angle ? How can you identify the greater and smaller angles ? What is a triangle ? Are all the triangles the same ? What is a quadrilateral ? Are all the quadrilaterals the same ? WARM-UP CHAPTER 1. LINES AND ANGLES Lesson Topics Pages 1.1 Review on Lines and Angles 9 1.2 Measuring Angles by Protractor 13 1.3 Types of Angles 21 1.4 Drawing Angles by Protractor 25 1.5 Problem on Angle 28 1.6 Review on Triangle 31 1.7 Measuring Sides of Triangle 36 1.8 Measuring Angles of Triangle 38 1.9 Problems on the Sum of Angles of Triangle 43 1.10 Review on Quadrilaterals 46 1.11 Measuring Angles of Quadrilateral 48 1.12 Problems on Sum of Angles of Quadrilateral 51 1.13 Drawing Perpendicular and Parallel Lines 54
Geometry 9 1.1 Review on Lines and Angles At the end of this topic, the students will be able to: ¾ recall the lines and angles. Learning Objectives Read, Think and Learn Point What is the shape and size of the tip of the sharpened pencil, needle, thorn, etc. They have no length, no breadth and no height. They are very tiny in shape and size. What are their tips called? They are just point. Point is denoted by dot (.). Line When a sheet of paper is folded and opened, what do you see? You will see the crease line on the folded part. This crease is like a line. folded paper unfold paper line (creasing) The edges of board, book, duster, ring, etc. are all like as lines. When two separate points are joined, what do you see? How many lines are drawn between them? There are so many line are drawn from these two points. The lines extend anywhere in continuous. point P M M M N N N
10 The Leading Mathematics - 5 So, they attach the arrow head on both ends. How many types of lines pass through these two points? A B P Q There are two types of lines. They are: (i) Straight line (ii) Curve Line The curve line is bending line that is not in the same way. But the straight line is not bend that is in the same way. P Q Curve line Circle and oval are also curve lines. A B Stretched wire and pole are like straight lines. Straight line How many types of straight lines are there? ray ray When a line is cut at a point, it forms two rays. A ray has an initial point and an arrow. CA and CB are rays. line segment When a line is cut at two points, the middle part that does not have arrow head is a line segment. It is a piece of line. CD is a line segment. A A A B C C C D D C C B B Angle Tell the name of the angular shapes that are found at your home, surroundings and school. Where do the angles form in this shapes?
Geometry 11 Read, Think and Learn Angle of Amount of Turn Fig. I Fig. II Fig. III Fig. IV Fig. V The watches have, hour and minute hands. They move around the watch from the fixed point. In fig (I) both second and hour hand are at the same place. In how many minutes does the minute hand in Fig (II) to Fig. (IV) reach? The minute hand moves a complete turn in 60 minutes or 1 hour as in Fig V. We say a complete turn has 360 equal parts. Each parts is called degree (°). This unit is used to measure the angle. So, a complete turn has 360° (read as 360 degree). 1 full turn = 360° (360 degree) An amount of turning of an object from the fixed point is called an angle. Points to be Remembered 1. A geometric shape that is used to indicate exact location in space that has no length, width, or thickness, is called a point. Point is denoted by dot (.). 2. A geometric shape that passes through at least two points is called a line. Arrow heads are used at the ends of the line. The lines are infinite in length. There are two types of lines; (i) Straight line and (ii) Curved line. 3. A half of a straight line is called a ray. 4. A piece of straight line is called a line segment. A line is formed when two points are joined straight. It has fixed length.
12 The Leading Mathematics - 5 5. When two lines or line segments intersect or join at a point, there forms an angle. The angle is denoted by ∠. 6. The capital letters of the English alphabet are used for naming the angle. It is written as ∠ABC or ∠CBA or ∠B. 7. The lines or line segments that form an angle, are called the arms. In the above angle, BA and BC are arms of the angle. 8. The point of intersection or joining point of two lines or line segments is called a vertex. In the above angle, B is the vertex. 9. The letter written between other two letters denotes the vertex. PRACTICE 1.1 Your mastery depends on practice. Practice as you play. Fill the correct answer in the given box, from the given angle. 1. What is the geometric name of the figure? 2. What is the name of the angle ? (Use 3 letters) 3. How do you read the symbols AB and AC ? (a) AB is (b) AC is 4. Which is the common point of the rays AB and AC ? 5. What is the special name of the common point A of the angle ∠BAC = ? 6. What do you call the rays in the given angle BAC? AB and AC 7. Name the angle BAC in another way. B C A A B C A B C Arm (Side) Arm(Side) Vertex
Geometry 13 1.2 Measuring Angles by Protractor At the end of this topic, the students will be able to: ¾ measure the angles by using protractor. Learning Objectives Read, Think and Learn What is the name of the diagram given below ? For what purpose is it used ? How can it be used to measure the angle ? 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 Outer semicircle (Outer scale) Inner semicircle (Inner scale) Base Line Centre Protractor The scale in the inner semicircle is used to measure anti-clockwise angle facing the right side and the outer scale is used to measure the clockwise angle facing the left side of the center. Steps of using protractor to measure an angle in degree Step 1: Fix the centre of the protractor on the vertex of an angle. Step 2: Line up the baseline of the protractor with one of the arms of the angle. Step 3: Use the scale on the protractor's base which starts at 0°. Clockwise Anti-clockwise
14 The Leading Mathematics - 5 See the following diagrams. Read the appropriate scale on the protractor and write the measure of angles in the boxes provided. 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 Anti-clockwise Angle Used inner sale Clock-wise Angle Used outer scale Be careful to measure the following types of angles. Write the measure of the following angles. (a) Very small angles 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180
Geometry 15 (b) Sides fall on both inner and outer scales. 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 (c) Do not confuse with the bottom line the and the baseline of protractor. 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 Bottom line of protractor Base line of protractor (d) When sides are very short (draw the dotted lines as shown below) 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 (e) Angles not multiple of 5 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 42°
16 The Leading Mathematics - 5 Example 1: Measure the given angle ∠AOB ? (a) What is the measure of ∠AOB ? (b) Find the value of a. Solution: (a) By measuring ∠AOB, we get ∠AOB = 90° (b) Again, ∠AOB = 6a or, 90° = 6a or, 90° 6 = a or, a = 15° Thus, the value of a = 15° Points to be Remembered 1. A semicircle device that is used for measuring angles is called protractor. 2. A inner scale protractor is used to measure anti-clockwise angle facing right side and the outer scale is used to measure clockwise angle facing left side of the center. 3. A protractor is used for measuring an angle in degree (°). 4. Steps of using protractor to measure an angle in degree; i. Fix the center of the protractor on the vertex of an angle. ii. Line up the baseline of the protractor with one of the arms of the angle. iii. Use the scale on the protractor's base which starts at 0°. PRACTICE 1.2 Your mastery depends on practice. Practice as you play. 1. Measure the size of the following angles by using a protractor. (a) (b) A O 6a B
Geometry 17 (c) (d) (e) (f) (g) (h) 2. Measure the angle and choose the correct answer. (a) (i) 10° (ii) 20° (iii) 90° (iv) 30° (v) 45° 3. Which of the angles below is 70°? Guess and check by measuring. (a) (b) (c) (b) (i) 30° (ii) 35° (iii) 40° (iv) 50° (v) 20°
18 The Leading Mathematics - 5 (d) (e) (f) 4. Observe the following figures and answer the following questions. (i) A C B (ii) O R P Q (a) How many angles are there in each figure? (b) Write the measure of each angle. (c) Write the relation between these angles. 5. Study the following figures and answer the following questions. (i) R OP Q (ii) OP Q R (iii) O Q R P (iv) O Q R P (a) How many angles are there in each figure? (b) Write the name of angles in each figure. (c) Write the measure of ∠POQ, ∠POR and ∠QOR in each figure.
Geometry 19 (d) What is the sum of ∠POR and ∠QOR? (e) Write the relation between ∠POQ, ∠POR and ∠QOR. 6. Observe the following figures and answer the following questions. (i) O A B C D (ii) O Q S P R (a) How many angles are formed in each figure ? Name them. (b) Measure all these angles. (c) Which angles have the same measurement in each figure? Write them. 7. Study the adjoining house-shaped plane figure. (a) Find the places where angles are formed in the picture and measure them. (b) Measure all these angles. (c) Which angles have the same measurement in each figure? Write them. 8. Study the adjoining open envelope. (a) How many angles are there in the open envelope ? (b) Count and name them. (c) Measure all these angles. 9. (i) Observe the adjoining figure. (a) Measure ∠ABC by using a protractor. (b) Find the value of 'P'. (c) Test the measure of ∠ABC. T A H O E P U S E N V L P O A B 3p C
20 The Leading Mathematics - 5 (ii) Study the given figure and answer the following questions. (a) Measure ∠PQR by using protractor. (b) Find the value of 'a'. (c) Test the measure of ∠PQR. (iii) Observe the given figure and answer the following questions. (a) Measure ∠ABC by using protractor. (b) Find the value of 'p'. (c) Test the measure of ∠ABC. (iv) Study the given figure and answer the following questions. (a) Measure ∠APF by using protractor. (b) Find the value of 'k'. (c) Test the measure of ∠APF. 1. Take a A4 paper and glue. Collect some bamboo sliced small sticks (sinka) or sulfur less match sticks or needle or such types of materials. Paste any two materials on A4 paper using glue to be formed an angle. Make six angles in different sizes. Name the angles and measure these all angles by using protractor and list them. Present it your classroom. 2. Draw any five angles on a A4 paper and measure these angles by using protractor. Demonstrate in your classroom. PROJECT WORK P Q 10a + 5° R A B 3p + 4° C A P 7k – 4° F
Geometry 21 1.3 Types of Angles At the end of this topic, the students will be able to: ¾ identify the types of angles according to their measures. Learning Objectives Read, Think and Learn Ö An angle of 90° is called a right angle. In the figure, ∠AOB = 90° is the right angle. Ö An angle less than 90° is called an acute angle. In the figure, ∠COD = 60° is an acute angle. Ö An angle less than 180° and greater than 90° is called an obtuse angle. In the figure, ∠XOY = 150° is an obtuse angle. Ö An angle of 180° is called a straight angle. In the figure, ∠POQ = 180° is the straight angle. Ö An angle greater than 180° is called a reflex angle. In the figure, ∠KOG = 240° is a reflex angle. Example 1: Observe the adjoining figure, (a) How many angles are formed in the given figure ? (b) Measure ∠AOC by using protractor. Write its measure. (c) Find the value of a. (d) What is the value of ∠BOC without measuring by a protractor? C 600 O O A 900 O B X 1500 Y B O P 1800 Q B O K 2400 G B O 5a 3a + 20° A O B C
22 The Leading Mathematics - 5 Solution: (a) 3 angles are formed in the given figure. (b) By measuring ∠AOC by using a protractor, we get ∠AOC = 80° (c) ∠AOC = 5a or, 80° = 5a or, 80° 5 = 16° or, a = 16° (d) We have, ∠AOB = 180° [ Measuring of straight angle] or, ∠AOC + ∠BOC = 180° [ Whole parts fact] or, 80° + ∠BOC = 180° or, ∠BOC = 180° – 80° \ ∠BOC = 100° Points to be Remembered 1. The angle which measures 90° is called a right angle. 2. The angle which measures less than 90° and greater than 0° is called an acute angle. 3. The angle which measures less than 180° and greater than 90° is called an obtuse angle. 4. The angle which measures 180° is called a straight angle. 5. The angle which measures greater than 180° is called a reflex angle. PRACTICE 1.3 Your mastery depends on practice. Practice as you play. 1. Observe the following figures and answer the following questions. (a) (b)
Geometry 23 (c) (d) i. Name the above angles. ii. What types of angles are they? Write in the above boxes. iii. Measure the angles and compare with your answer. 2. Classify the following angels by using the words acute, right, obtuse, straight and reflex. 110°, 18°, 7°, 181°, 90°, 270°, 360°, 79°, 20°, 246° 3. How many angles you can see in the figure? (a) Name the acute angles. (b) Name the angle that is right. (c) Name the obtuse angles. (d) Name the reflex angles. 4. (a) Name any two acute angles that you can see in the adjoining figure. (b) Name any two obtuse angles that you can see in the adjoining figure. A B C O D A D B C
24 The Leading Mathematics - 5 5. The hands of a clock make a right angle at 9'o clock. (a) Write the time when the hands of the clock make another right angle. (b) Write the time in whole number when the hands of the clock make an acute angle. (c) Write the time in whole number when the hands of a clock make an obtuse angle. 6. Observe the given figure. (a) How many angles are there in the figure? (b) Measure the angle ABC and ABD. Write their measures. (c) Find the value of 'a'. (d) Find the measure of ∠CBD without measuring by a protractor. 7. Observe the given figure, (a) How many angle are there in the figure? (b) Measure the angle QOR by using protractor and write its measure. (c) Find the value of a. (d) Find the measure of ∠POR without measuring by a protractor. A B C D a P O Q 3a R
Geometry 25 1.4 Drawing Angles by Protractor At the end of this topic, the students will be able to: ¾ draw the angles of the given measures by using protractor. Learning Objectives Read, Think and Learn Draw an angle of 30° at the given point O on a given ray AB. Given point. A O B Step 1: Place the centre of the protractor at O. Step 2: Adjust the base line of the protractor along AB so that the centre is on the point O. A O B 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 C Fig (I) Step 3: Read the scale 0°, 10°, 20°, 30° etc. on the protractor until you reach the required degree of the angle. Then mark with a sharp pencil at 30° and name C. (i) If you read the degrees on the right hand side of O, you are drawing anticlockwise angle of 30°. You indicate the direction as shown in fig (i). Using inner scale for anti-clockwise 30°
26 The Leading Mathematics - 5 (ii) If you read the degrees on the left hand side of O, you are drawing clockwise angle of 30°. You indicate the direction as shown below in fig (ii). A X B 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 C Fig II (iii) Lift the protractor and joint the marked point at 30° and the point O by using its straight edge or a scale. (iv) Then the drawn angle is either ∠BOC or ∠AOC. Points to be Remembered Steps of drawing an angle of 50°; 1. Draw a line segment AB and mark a point O on it. 2. Place the center of the protractor at O. 3. Adjust the baseline of the protractor along AB so that the center is on the point O. 4. Read the outer scale 0°, 10°, 20°, 30° etc. on the protractor until you reach the required degree of the angle 50°. Then mark with a sharp pencil at 50° and name C. 5. Join C with O and form the angle BOC. Hence, ∠BOC = 50°. PRACTICE 1.4 Your mastery depends on practice. Practice as you play. 1. Draw the following angles by using the inner scale of protractor for anticlock-wise angles. a. 10° b. 20° c. 50° d. 65° e. 35° f. 70° g. 120° h. 90° i. 180° j. 165° k. 155° l. 135° clockwise 30° Angle Used inner scale
Geometry 27 2. Draw the following angles by using the outer scale of the protractor for clockwise angle. a. 40° b. 30° c. 70° d. 25° e. 45° f. 80° g. 90° h. 120° i. 170° j. 180° k. 165° l. 150° 3. With the help of a protractor, draw the angles similar to the following angles. 4. Draw a line segment XY. Draw an anti-clockwise angle of 30° at X and Y. 5. Draw a line segment PQ. Draw a clockwise angle of 60° at both P and Q. 6. Draw a line segment AB. Draw an anti-clockwise angle of 30° at A and a clockwise angle of 60° at B. Produce the arms when they interest. What type of figure do you see ? Name it. (a) (b) (c) (d) Project Work Draw a line segment AB = 6 cm on a sheet of paper and draw the angles 90° at both A and B. Take the points on the arms of both angles D and C such that AD = BC = 5 cm respectively. Join C and D. What type of figure do you see? Name it. Also, name C and D. What do you find ?
28 The Leading Mathematics - 5 1.5 Problems on Angle Read, Think and Learn ∠AOC is a right angle. What is the measure of ∠AOC? Draw a line OB in ∠AOC such that ∠AOB = 30° What is the measure of ∠BOC ? For this, ∠BOC + ∠AOB = ∠AOC [By whole parts] or, ∠BOC + 30° = 90° or, ∠BOC = 90° – 30° = 60° ∴ ∠BOC = 60° CLASSWORK EXAMPLES Example 1: Observe the given figure and answer the following questions. (a) Find the value of 'a'. (b) Find the measure of ∠AOC and ∠BOC. (c) Test the sum of ∠AOC and ∠BOC is 180°. Solution: (a) Here, ∠AOB = 180°, ∠AOC = 3a + 20°, ∠BOC = 5a Now, we know that ∠AOC + ∠BOC = ∠AOB or, 3a + 20° + 5a = 180° or, 8a = 180° – 20° or, a = 160° 8 or, a = 20°. (b) ∠AOC = 3a + 20° = 3 × 20° + 20° = 80°. (c) ∠BOC = 5a = 5 × 20° = 100°. O A B C 60° 30° 3a + 20° 5a A O B C
Geometry 29 Points to be Remembered 1. The sum of angles at a point made on the same side of a straight line is 180°. 2. The sum of angles at a vertex of a right angle on its same side is 90°. PRACTICE 1.5 Your mastery depends on practice. Practice as you play. 1. Observe the adjoining figure. (b) If ∠ABC = 60° in the given figure, find the value of p. (c) What is the complementary angle of 60°? 2. Observe the adjoining figure. (a) If ∠PQR = 65°, in the adjoining figure, find the value of a. (b) What is the supplementary angle of 65°? 3. Find the value of 'a' in each right angle. (a) 5a B C A (b) 65° + a Q P R (c) Y X 3a + 21° Z 4. Observe the following figures. (a) a 65° R Q P S (b) 35° a Q R P S (c) 4a + 10° 40° R Q P S 3p B C A 10a + 5° Q R P
30 The Leading Mathematics - 5 i. Name the greatest angle in the above each figures and write its measure. ii. Find the value of 'a'. iii. Find the measure of ∠SQR. 5. Observe the given figures below. (a) A O B 3a (b) A O B 9a (c) O B A 6a – 12° i. Write the measure of ∠AOB. ii. Find the value of 'a' in each straight angle. 6. Observe the following figures. (a) O D E C a 135° (b) C A O B 111° 3a + 15° (c) S P Q R a + 5° 7a + 15° i. Find value of 'a'. ii. Find the measure of unknown angles. 7. Observe the given picture and answer the following questions. (a) What is the measure of ∠ABE? (b) How much angle made by the ladder with the ground? Find it. D A C E B 125°
Geometry 31 1.6 Review on Triangle Read, Think and Learn Tell the name of the triangle-shaped objects that are found at your home, surroundings and school. What are the general shapes of these objects? Draw them. angles or corners B C side side side A Vertex Vertex Vertex Ö What is the figure called ..................................... Ö How many sides does the polygon have? ..................................... Ö How many angles does the polygon have? ..................................... A polygon with three sides and three angles is called a triangle. The symbol "∆" denotes a triangle. The above triangle is denoted by '∆ABC' read as triangle ABC. The same triangle can be denoted by writing symbol ∆ and three vertices in any order such as ∆BAC of a specific angle. But the vertex ∆CAB or ∆ACB, etc. is always in the middle. Notation: The side opposite to the given angle is named by a capital letter and denoted by the corresponding small letter. Example: The opposite side of ∠B is denoted by b. Hence, the sides of ∆ABC are a, b, c.
32 The Leading Mathematics - 5 Points to be Remembered 1. A closed geometric figure made by three sides is called a triangle. The symbol "∆" denotes a triangle. A triangle ABC is denoted by ∆ABC or ∆BCA or ∆CAB or ∆BAC or …. 2. A triangle has three vertices and three angles. The vertices of the triangle are always denoted by the capital letters of the English alphabet and its angles, corresponding small letters in general. 3. The side opposite to the given angle named by capital letter is denoted by the corresponding small letter of the angle. The opposite side of ∠B of ∆ABC is denoted by AC or b. Hence, the sides of ∆ABC are BC = a, AC = b, AB = c. 4. In triangle, the opposite part of a vertex or angles is the side and the opposite part of the side is the vertex or angle. PRACTICE 1.6 Your mastery depends on practice. Practice as you play. 1. Fill in the blanks. (a) How many sides are there in the given polygon? .................... (b) This polygon is called a .................... (c) How many vertices does the triangle have? .................... and what are they ? ...................., ...................., .................... . (d) Write the name of the triangle by using symbol '∆'. .................... (e) Write the name of the sides of the ∆ABC. The sides are ...................., ...................., .................... (f) The sides XY and XZ form the angle .................... (g) The angle Y is formed by the sides .................... and .................... (h) The side opposite to the angle Y is .................... and is denoted by .................... X Y Z
Geometry 33 (i) The opposite angle of the side XY is .................... (j) Name the ∆XYZ in one of the other five ways. .................... 2. Look at the triangle and answer the following questions. (a) How many vertices does it have? ........................ (b) What are its vertices ? ........................ (c) Can you name this triangle? ........................ (d) Can you write the name of this triangle? .................... (e) The opposite side of the angle P is ........................ (f) The opposite angle of the side PQ is ........................ (g) The angle formed by the sides PR and RQ is ........................ 3. Copy the triangle on your square grid. 4. Do all the activities for the following triangles as in question 2. (a) P Q R (b) C D E (c) K L N Q R P (a) Name the vertices of your own. (b) How do you name it? ................................. (c) Name this triangle in the rest of the five remaining ways. (d) Write two sides of the triangle and their included angle. Draw any three triangles with different names on an A4 paper and measure their interior angles by using a protractor. Find their sum in each triangle. Show your project work in the classroom. PROJECT WORK
34 The Leading Mathematics - 5 1.7 Measuring Sides of Triangle At the end of this topic, the students will be able to: ¾ measure the sides of a triangle. Learning Objectives Read, Think and Learn Look at the centimeter ruler given below. It is used to measure the length of a segment in centimeter and millimeter. Answer the following questions: 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 How do you read the number 1 on the scale? 1 centimeter (cm) or 10 millimeter (mm) Write down the length shown on the rulers. 0 1 2 3 4 5 A B C A B C 0 1 2 3 4 5 0 1 2 3 4 5 A B C 10 mm= 1 cm or, 1 cm = 10 mm mm is the small non-numbered line segment unit.
Geometry 35 Discuss which are easy to measure the length? The unit of measuring a line segment is Centimetre (or cm). Steps of measuring sides of triangles Step 1: Place the starting point of the scale at the left end of the given segment. Step 2: Line up the straight edge of the scale with given line segment. Step 3: Use the scale on the straight edge which starts at 0 at one end and read the number of centimeters or millimeters up to the next end of the segment. Follow the steps of measuring segments to measure the length of segments AB, PQ, XY and MN given above. Estimate the length of each segment, then measure the length and compare it with your estimation. A B B C C A e.g. For line segment AB, i. Estimate - 2.5 cm ii. Measure - 3 cm iii. Compare - difference of 5 mm Add end to end of the segments AB, BC and CA. What did you find? Measure the length of the sides of the following triangles. (a) Y X Z (b) A B C (c) E F D XY = ....., YZ = ....., ZX = ..... ................................ ...................................
36 The Leading Mathematics - 5 Ö Add the lengths of any two sides of each triangle and compare with the third side. Ö Tell the main fact of the triangle. Points to be Remembered 1. Steps of measuring sides of triangles; (i) Place the starting point of the scale at the left end of the given segment. (ii) Line up the straight edge of the scale with given line segment. (iii) Use the scale on the straight edge which starts at 0 at one end and read the number of centimeters or millimeters up to the next end of the segment. PRACTICE 1.7 Your mastery depends on practice. Practice as you play. 1. Use a centimeter scale to measure the side of the following triangles. (a) (b) (c) 2. Measure the length of sides of the following triangles by using the cm scale. Classify the triangles and find their perimeter. S.N. Triangles Measure of side Sum of two sides Comparing with the third side (a) P Q R PQ = ......... QR = ......... PR = ......... PQ + QR = ..... PQ + PR = ..... QR + PR = ..... PQ + QR ..... PR PQ + PR .... QR QR + PR .... PQ
Geometry 37 (b) F D E (c) K M N (d) A B C (e) M N O (f) Z X Y
38 The Leading Mathematics - 5 1.8 Measuring Angles of Triangle At the end of this topic, the students will be able to: ¾ measure the angles of a triangle. Learning Objectives Read, Think and Learn Look at the diagram below. (a) What are the size of the angles in the triangle ABC? Write your answers and name of the angle in boxes provided. C 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 A B In degree Types of angle. ∠BAC = ∠ABC = ∠ACB = ∠A is equal to 90°. What about the other two angles? The remaining angles are acute angles.
Geometry 39 Measure all the angles of DABC and fill the results in the table below: Angle ∠BAC ∠ABC ∠ACB ∠BAC + ∠ABC + ∠ACB Measurement ............... ............... ............... ............... What is the sum of the measurement of the angles in ∆ABC ? Compare with your friend's result. Is the sum the same? Consult with your other friends as well. What can we conclude from this? The sum of the measurement of the angles of a triangle is 180°. (b) What are the size of the angles in the triangle XYZ ? Write the sizes and types of the angles of ∆XYZ. X Y 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 Z In degree Types of angle ∠YXZ = ∠XYZ = ∠XZY =
40 The Leading Mathematics - 5 In DXYZ, ∠X = 120° which is obtuse. What about the other two remaining angles? They are acute angles. Measure all the angles of DXYZ and fill the results in the table below: Angle ∠YXZ ∠XYZ ∠XZY ∠YXZ + ∠XYZ + ∠XZY Measurement ............... ............... ............... ............... What is the sum of the measurement of the angles in ∆XYZ ? Compare with your friend's result. Is the sum the same ? Consult with your other friends as well. What can we conclude from this? (c) Measure the angles of ∆PQR. Type of angle ∠PQR = = ∠QPR = = ∠PRQ = = P Q R 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 110 120 130 100 90 80 0 0 180 Measure all angles of DPQR and fill the results in the table below: Angle ∠QPR ∠PQR ∠PRQ ∠QPR + ∠PQR + ∠PRQ Measurement ............... ............... ............... ...............
Geometry 41 What is the sum of the measurement of the angles in ∆PQR ? Compare with your friend's result. Is the sum the same ? Consult with your other friends as well. What can we conclude from this? PRACTICE 1.8 Your mastery depends on practice. Practice as you play. 1. Measure the angles of the following triangles. Find the sum of all the three angles of each triangle. X Y Z Angle Measurement ∠YXZ ............ ∠XYZ ............ ∠XZY ............ ∠YXZ + ∠XYZ + ∠XZY ............ (a) (b) Angle Measurement ∠QPT ............ ∠PQT ............ ∠PTQ ............ ∠QPT + ∠PQT + ∠PTQ ............ (c) Q P T Angle Measurement ∠CTV ............ ∠TCV ............ ∠TVC ............ ∠CTV + ∠TCV + ∠TVC ............ C T V
42 The Leading Mathematics - 5 (d) (d) Project Work Draw a triangle ABC on an A4 paper. Cutout it from the paper. Cut its angles as shown in the adjoining figure. Join all the angles at the same point. What do you see? Write your conclusion. Prepare a report about it and present in it your classroom. A B C Angle Measurement ∠RMP ............ ∠MRP ............ ∠RPM ............ ∠RMP + ∠MRP + ∠RPM ............ M P R Angle Measurement ∠QPR ............ ∠PQR ............ ∠PRQ ............ ∠QPR + ∠PQR + ∠PRQ ............ Q P R
Geometry 43 1.9 Problems on the Sum of Angles of Triangle Read, Think and Learn In the adjoining figure, ABC is a triangle. What is the sum of angles in ∆ABC? If the measures of ∠BAC and ∠AOC are 50° and 65° respectively, how can we find the measure of ∠ACB? For this, we know that ∠BAC + ∠ABC + ∠ACB = 180° [Being the measure of straight angle] or, 50° + 65° + ∠ACB = 180° or, ∠ACB = 180° – 115° or, ∠ACB = 65° Hence, the measure of ∠ACB is 65°. Example 1: In ∆PQR, ∠PRQ = 45°, ∠QPR = 2a and ∠PQR = a. (a) Find the actual measure of ∠PQR. (b) Find the actual measure of ∠QPR. Solution: (a) Here, in ∆PQR, ∠PRQ = 45°, ∠QPR = 2a, ∠PQR = a, Now, we know that, ∠PRQ + ∠QPR + ∠PQR = 180° or, 45° + 2a + a = 180° or, 3a = 180° – 45° or, a = 135° 3 or, a = 45° ∴ ∠QPR = 45° (b) ∠QPR = 2a = 2 × 45° = 90° Hence, the actual measure of ∠QPR is 90°. 65° 50° ? A B C a 2a 45° P Q R
44 The Leading Mathematics - 5 PRACTICE 1.9 Your mastery depends on practice. Practice as you play. 1. In ∆ ABC, (a) If ∠A = 30° and ∠B = 68°, find ∠C. (b) If ∠A = 48° and ∠C = 25°, find ∠B. 2. Observe the following triangles and answer the following questions. (a) Find the value of 'a' in each triangle. (b) Find the measure of the unknown angles. i. 67° 48° a P Q R ii. 65° 55° a° A B C iii. U S T 30° a iv. 25° 28° a + 5° X Y Z v. E F D 2a 35° 3a vi. G H I a 30° 2a vii. 40° a K G D viii. 65° M R S a + 5° ix. 32° 38° a M N L x. 55° 55° 3a + 10° W X Y A B C
Geometry 45 xi. R P S 33° 5x 2x xii. 32° x H M B 3. (a) The two angles of a triangle are 45° and 53°. Find the measure of its third angle. (b) What is the measure of the remaining angle of a triangle having two angles 80° and 28°? 4. (a) If one angle of a right-angled triangle is 42°, find its third angle. (b) If one angle of a right-angled triangle is 58°, find its third angle. 5. Find the measure of ∠PQR in each triangle. (a) P 60° 36° 3a Q R (b) P 48° 40° 4p° R Q (c) P Q R 90° 2q 2q + 14° (d) P Q 85° 2c – 10° R 3c + 5° (e) Q P R 75° 2b + 10° 3b – 5° (f) Q P R 3a + 25° 35° 2a – 15°
46 The Leading Mathematics - 5 1.10 Review on Quadrilaterals At the end of this topic, the students will be able to: ¾ recall the quadrilateral and its parts. Learning Objectives Read, Think and Learn Name the quadrilateral-shaped objects that are found at your home, surroundings and school. What are the general shapes of these objectives. Draw them. From the adjoining drawn, Write the answers of the following questions. How many sides does it have? .......................... How many angles does it have? .......................... This polygon has 4 sides and 4 angles. A polygon having 4 sides and 4 angles is called a quadrilateral. Opposite sides Opposite angles AB and CD: AD and BC ∠A and ∠C; ∠B and ∠D Side AB and AD make BAD at A. Sid BA and BC make ∠ABC at B. Side CB and CD make ∠BCD at C. Side DC and DA make ∠ADC at D. Activity Ö Name the vertices by capital letters. Ö Name the quadrilateral. Ö Name the sides. Ö Name the angles. Ö Write 2 pairs of opposite sides. Ö Write 2 pairs of opposite angles. A B C D
Geometry 47 Do the same activity in 1 for the following quadrilaterals. PRACTICE 1.10 Your mastery depends on practice. Practice as you play. 1. Write down the sides, angles, pair of opposite angles and pair of opposite sides of each quadrilateral. S.N Quadrilateral Sides Opposite sides Angles Opposite angles (a) A B C D (b) Q P S R (c) W Z Y X (d) G F D E (e) I L J K
48 The Leading Mathematics - 5 1.11 Measuring Angles of Quadrilateral At the end of this topic, the students will be able to: ¾ measure the angles of quadrilateral. Learning Objectives Read, Think and Learn I have a quadrilateral ABCD. How can I get its angular sum? It is so easy. Measure its angles and add them. X W Z Y X W Z Y 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 10 170 20 160 30 150 40 140 50 130 60 120 70 110 100 90 80 70 60 50 40 30 20 10 170 180 160 150 140 130 120 110 100 90 80 0 0 180 ∠WXY = 90° ∠XWZ = 90° ∠WZY = 120° ∠XYZ = 60° Total = 360°
Geometry 49 Measure the angles ∠XWZ, ∠WXY, ∠XYZ, and ∠YZW. Fill the result in the table below: Angles ∠XWZ ∠WXY ∠XYZ ∠YZW Sum Measurement Hence, the sum of angles of a quadrilateral is 360° or 4 right angles. It forms a complete angle. PRACTICE 1.11 Your mastery depends on practice. Practice as you play. 1. Observe the following plane figures. (a) Measure the size of the angles of each figure. (b) Write the pair of opposite angles of each quadrilateral. (c) Find the sum of opposite angles in each figure. (d) Find the total of all the angles in each figure. i. ii. iii. iv. v. vi. P Q S R L M O N D C B A E F G H S T U F K L M J
50 The Leading Mathematics - 5 2. Measure the angles of a quadrilateral HIRA. Also, measure the angles in ∆HAR and ∆HIR and then fill in the table below: In quad. HIRA ∠H ∠I ∠R ∠A Sum Measure Write the conclusion. 3. Observe the quadrilateral PREM and write the answers of the following questions. Measure the angles of the triangles PEM and PRE. Fill the result in the table below. In ∆PEM ∠PEM ∠EMP ∠MPE ∠PRE ∠REP ∠EPR Sum Measure Write the conclusion. Draw any three quadrilaterals with different names on an A4 paper and measure their interior angles by using a protractor. Find their sum in each quadrilateral. Show your project work in the classroom. PROJECT WORK R I H A M P R E