Preface. It is hoped that this book will help students to gain confidence in the subject and be better equipped to face the examinations.

Transcription

1 Preface Secondary 1 Mathematics Tutorial 1 and 1 are designed to prepare Secondary 1 students in their understanding and application of mathematical concepts, skills and processes. What s covered in this book? Written in accordance with the latest syllabus, each chapter includes Objectives, Key oncepts and Formulae and Worked xamples to supplement and complement the lessons taught in school. Practice questions are structured as core, consolidation and challenging to ensure steady improvement and quick mastery of concepts. ooks 1 and 1 cover all the topics for the entire school year. dditional feature that provides important notes, tips, examples and common student errors. Important concepts are highlighted to enhance understanding and retention. Tests and ssessments There are 4 assessment papers: Mid Year xamination Paper 1 and 2 (found in ook 1) and nd of Year xamination Paper 1 and 2 (found in ook 1). nswer Key Fully worked solutions are provided for students to understand better how each problem is solved. These also serve as a tool for self study and assessment. It is hoped that this book will help students to gain confidence in the subject and be better equipped to face the examinations. The ditorial Team

2 ontents ontinuation from Secondary 1 Mathematics Tutorial 1 (hapters 1 7) hapter Objectives Page Lines, ngles and Triangles pply properties of vertically opposite angles, angles on a straight line and angles at a point pply properties of interior angles, corresponding angles and alternate angles formed by parallel lines Solve problems involving angles in triangles and quadrilaterals Polygons and Geometrical onstruction pply properties of a polygon to solve geometrical problems onstruct angle bisectors and perpendicular bisectors onstruct triangles and quadrilaterals Percentage xpress one quantity as a percentage of another quantity ompare two quantities in terms of percentage alculate percentage increase and decrease in quantities Solve problems involving reverse percentages Solve problems involving discounts and GST oordinates and Linear Functions State the coordinates of a point on a artesian oordinate system Plot the graph of linear functions Find the gradient of a linear graph rea and Perimeter of Plane Figures alculate the area of squares, rectangles, parallelograms, trapeziums and circles alculate the area and perimeter of composite figures Volume and Surface rea of Solids alculate volume and surface area of cubes, cuboids, prisms and cylinders Solve problems involving volume and surface area of composite solids Statistics onstruct and interpret tables, bar graphs, pictograms, line graphs and pie charts xplain why certain statistical diagrams can lead to misinterpretation of data nd of Year xamination Paper nd of Year xamination Paper Fully worked solutions S1 S53

3 hapter 8 Objectives Lines, ngles and Triangles pply properties of vertically opposite angles, angles on a straight line and angles at a point pply properties of interior angles, corresponding angles and alternate angles formed by parallel lines Solve problems involving angles in triangles and quadrilaterals Key oncepts and Formulae 1 Types of triangles Right triangle Isosceles triangle quilateral triangle Scalene triangle 60º 2 sides are perpendicular a a 2 sides are equal; base s are equal 60º 60º ll 3 sides are equal ll 3 sides are unequal 2 Types of angles acute angle (between 0º and 90º) 3 ngles on a straight line obtuse angle (more than 90º but less than 180º) reflex angle (more than 180º) bº + bº = 180º ( s on a st. line) 4 ngles at a point dº bº + bº + + dº = 360º ( s at a point ) 5 Vertically opposite angles b a c d a = c b = d (vert. opp. s) (vert. opp. s) Singapore sia Publishers Pte Ltd hapter 8 Lines, ngles and Triangles

4 6 ngles formed by parallel lines orresponding angles Interior angles lternate angles eº bº dº f º = bº (corr. s, // ) + dº = 180º (int. s, // ) eº = f º (alt. s, // ) 7 ngles in a Triangle bº + bº + = 180º ( sum in ) bº = + bº (ext. of ) 8 ngles in a Quadrilateral bº bº dº dº + bº + + dº = 360º ( sum in quad.) = and bº = dº (opp. s of //ogram) 9 iagonals of rhombus, square and kite intersect at 90º rhombus square kite Singapore sia Publishers Pte Ltd hapter 8 Lines, ngles and Triangles

5 WORKe exmple 1 In the figure, and are straight lines. 2 5 F Only assume lines are straight if the question states so. Find (a) the value of x, (b), (c) obtuse. Solution: (a) = 180º ( s on a st. line) 8 = 180º x = 22.5 (b) = (vert. opp. s) = 2 = 2(22.5º) = 45º When an unknown angle is denoted with its degree, e.g., then x represents a number i.e. x itself has no unit. (c) = 180º ( s on a st. line) = 180º 45º = 135º When stating reasons, you may use abbreviations Singapore sia Publishers Pte Ltd 3 hapter 8 Lines, ngles and Triangles

6 WORKe exmple 2 In the figure, is a parallelogram and is a straight line. Find the values of x, y and z. 29º 38º Solution: yº zº You need not start solving for x first. zº = (alt. s, //) = 29º + 38º = 67º yº = (alt. s, //) = 38º + 29º + 38º = 180º ( sum in ) x + 67º = 180º = 113º x = 113, y = 38 and z = 67 WORKe exmple 3 In the figure, //. Find. 31º lternatively, = 180º 31º 64º = 85º = 180º 85º = 95º Solution: = 31º + 64º = 95º = 180º 95º = 85º 64º (ext. of ) (int. s, //) Singapore sia Publishers Pte Ltd 4 hapter 8 Lines, ngles and Triangles

7 WORKe exmple 4 In the diagram below, find (a) the value of x, (b) the reflex F. 290º obtuse angle reflex angle F º F Solution: (a) raw a line such that ////F. = = 180º (2 + 70º) = 180º 2 70º = 110º 2 290º º F (alt. s, //) (int. s, //F) 290º + + = 360º ( s at a pt.) 290º º 2 = 360º 400º = 360º = 40º x = 40 rawing an additional line is a common technique used in Geometry. (b) Substituting x = 40 in º, Obtuse F = 2(40º) + 70º = 150º Reflex F = 360º 150º = 210º ( s at a pt.) n angle zº is reflex, if 180º < zº < 360º. Singapore sia Publishers Pte Ltd 5 hapter 8 Lines, ngles and Triangles

8 core Practice ngles on a Straight Line 1 Given that O is a straight line, find the value of x in each of the following figures. Sum of s on a straight line = 180º. ( s on a st. line) Observe that x is a number with no units. (a) (b) 140º O 70º O (c) (d) 2 3 O O + 16º 2 28º Singapore sia Publishers Pte Ltd 6 hapter 8 Lines, ngles and Triangles