1A_Ch6(1)

1A_Ch6(2) 6.1Basic Geometric Knowledge A Points, Lines and Planes B Angles C Parallel and Perpendicular Lines Index

1A_Ch6(3) 6.2Plane Figures A Circles B Triangles C Polygons Index Introduction to Plane Figures

1A_Ch6(4) 6.3Three-dimensional Figures A Introduction B Sketch the Two-dimensional (2-D) Representation of Simple Solids Index

1A_Ch6(5) 6.4Polyhedra A Introduction to Polyhedra B Making Models of Polyhedra Index

Points, Lines and Planes 1.Refer to the right figure. Index A) 1A_Ch6(6) A CD EB i.A is a point. ii.BE is a line. iii.CD is a line segment, C and D are called the end points of that line segment. iv.Figure ACD represents a plane. 6.1Basic Geometric Knowledge

Points, Lines and Planes 2.Relations among Points, Lines and Planes i.The straight line in the common part of two planes is called the line of intersection. ii.The two lines meet each other at a point, that point is called the point of intersection. Index A) 1A_Ch6(7) line point of intersection line 6.1Basic Geometric Knowledge + ExampleExample + Index 6.1Index 6.1

(a)Name all the line segments and planes in the given figure. (b)Which point is the point of intersection of MQ and PN ? (a)Line segments : Index 1A_Ch6(8) M N P Q O MN, MO, NO, OP, OQ, PQ, MQ, NP Planes : MNO, OPQ 6.1Basic Geometric Knowledge (b)Point of intersection of MQ and PN : O + Key Concept 6.1.1Key Concept 6.1.1

Types of Angles Angles can be classified according to their ‘sizes’ as follows: Index B) 1A_Ch6(9) Note : In a figure, a right angle is usually indicated by the symbol ‘ ’ but not an arc ‘ ’. Acute angle Right angle Obtuse angle Straight angle Reflex angle Round angle 6.1Basic Geometric Knowledge + ExampleExample + Index 6.1Index 6.1

What kind of angle is each of the following angles in the given figure? (a) ∠ AOB(b) ∠ BOD (a) ∠ AOB = 180° Index 1A_Ch6(10) A O B C D 140° 180 ° ∴ ∠ AOB is a straight angle. (b) ∠ BOD= ∠ BOC – ∠ COD = 140° – 90° = 50° ∴ ∠ BOD is an acute angle. 6.1Basic Geometric Knowledge

Index 1A_Ch6(11) What kind of angle is each of the following angles in the given figure? (a) ∠ AFE (b) ∠ AHD (c) ∠ EFB (a) ∠ AFE = 120° ∴ ∠ AFE is an obtuse angle. (b) ∠ AHD = 90° ∴ ∠ AHD is a right angle. (c) ∠ EFB= ∠ EFA + ∠ AFB = 120° + 60° = 180° ∴ ∠ EFB is a straight angle. Fulfill Exercise Objective  Classify an angle. 6.1Basic Geometric Knowledge + Key Concept 6.1.2Key Concept 6.1.2

Parallel and Perpendicular Lines i.RS and TU are a pair of parallel lines. We can write RS // TU. ii.AB and TU are a pair of perpendicular lines. We can write AB ⊥ TU. Index C) 1A_Ch6(12) R S T U B A iii.Parallel and perpendicular lines can be constructed by a ruler and a set square. 6.1Basic Geometric Knowledge + ExampleExample + Index 6.1Index 6.1

Name all the parallel lines and perpendicular lines in the given figure. Index 1A_Ch6(13) Parallel lines : AG // DC // FE, GF // DE Perpendicular lines : AB ⊥ BC, AG ⊥ GF, GF ⊥ FE, FE ⊥ ED, ED ⊥ DC 6.1Basic Geometric Knowledge + Key Concept 6.1.3Key Concept A G D F E C B

Introduction to Plane Figures 1.A geometric figure formed by points, lines and planes lying in the same plane is called a plane figure. Index 1A_Ch6(14) E.g. Circle Triangle Polygon 6.2Plane Figures + Index 6.2Index 6.2

1.O is the centre. 2.OP is the radius. 3.AOB is the diameter. 4.The curve AQBPA which forms the entire circle is the circumference. 5.The curve AP is part of the circumference, called an arc of the circle. 6.Circles and arcs can be constructed by a pair of compasses. Index 1A_Ch6(15) O AB P Q A) Circles 6.2Plane Figures

Note : i.‘Circumference’, ‘radius’ and ‘diameter’ can represent lengths as well. ii.Diameter = 2 × radius Index 1A_Ch6(16) arc diameter radius A) Circles 6.2Plane Figures + ExampleExample + Index 6.2Index 6.2

It is known that O is the centre of each of the following circles, find the values of the unknowns. (a)(b) Index 1A_Ch6(17) x cm 12 cm O O y m 4.5 m (a)Diameter = 12 cm ∴ x= 12 ÷ 2 = 6 (b)Radius = 4.5 m ∴ y= 4.5 × 2 = 9 6.2Plane Figures + Key Concept 6.2.2Key Concept 6.2.2

1.In the above triangle, Index 1A_Ch6(18) B) Triangles A B C 2.The sum of the three angles of a triangle is 180°. i.the line segments AB, BC and CA are called the sides of △ ABC, ii.points A, B, C are called the vertices (singular : vertex) of △ ABC. 6.2Plane Figures + ExampleExample

3.Classification of triangles : Index 1A_Ch6(19) B) Triangles Acute-angled triangle Right-angled triangle Obtuse-angled triangle 6.2Plane Figures

3.Classification of triangles : Index 1A_Ch6(20) B) Triangles Scalene triangle Isosceles triangle Equilateral triangle 6.2Plane Figures + ExampleExample

4.Triangles can be constructed by a protractor and a pair of compasses etc. according to given conditions: i.Given three sides of a triangle. ii.Given two sides and the included angle of a triangle. Index 1A_Ch6(21) B) Triangles 6.2Plane Figures + ExampleExample + Index 6.2Index 6.2

Index 1A_Ch6(22) Find the unknown angle a in the figure. a + 120° + 40° = 180° a + 160° = 180° a = 180° – 160° = 20° 6.2Plane Figures

Index 1A_Ch6(23) Find the unknowns x and y in △ ABC as shown. In △ ABD, x + 62° + 90° = 180° x + 152° = 180° x = 180° – 152° = 28° In △ ABC, 28° + 62° + 48° + y = 180° 138° + y = 180° y = 180° – 138° = 42° Fulfill Exercise Objective  Find an unknown angle in a triangle. 6.2Plane Figures + Key Concept 6.2.3Key Concept 6.2.3

For the above triangles A, B, C and D, identify (a)scalene obtuse-angled triangle? (b)isosceles acute-angled triangle? Index 1A_Ch6(24) (a)C (b)D ABCD 6.2Plane Figures + Key Concept 6.2.4Key Concept 6.2.4

Index 1A_Ch6(25) Construct △ ABC, where AB = 4 cm, BC = 3 cm and AC = 3.5 cm. Steps : 1.Use a ruler to draw a line segment AB of 4 length cm. 2.With centre at A and radius 3.5 cm, use a pair of compasses to draw an arc. 3.With centre at B and radius 3 cm, use a pair of compasses to draw another arc. 4.The two arcs drawn should meet at C. 5.Join AC, then BC. △ ABC is drawn. Fulfill Exercise Objective  Construct a triangle. 6.2Plane Figures

Index 1A_Ch6(26) Construct △ PQR, where PQ = 3 cm, ∠ RPQ = 50° and RP = 4 cm. Fulfill Exercise Objective  Construct a triangle. Steps : 1.Use a ruler to draw a line segment PQ of length 3 cm. 2.Use a protractor to draw ∠ TPQ that measures 50°. 3. Use a ruler to mark a point R on PT produced such that RP = 4 cm. 4.Join QR, then △ PQR is drawn. 6.2Plane Figures + Key Concept 6.2.6Key Concept 6.2.6

1.A plane figure formed by 3 or more line segments is called a polygon. 2.A polygon is usually named by the number of its sides or n-sided polygon (n is whole number). Index 1A_Ch6(27) C) Polygons 6.2Plane Figures

Index 1A_Ch6(28) 3.The line segments that form a polygon are called sides of the polygon. 4.The point where two adjacent sides meet is called a vertex of the polygon. 5.The line segment joining two non-adjacent vertices is called a diagonal. C) Polygons diagonal vertex side 6.2Plane Figures

Index 1A_Ch6(29) Classification of polygons : C) Polygons Equilateral polygon Equiangular polygon Regular polygon 6.2Plane Figures + Index 6.2Index ExampleExample

Index 1A_Ch6(30) (a) A, C (b) A, B (c) A For each of the following polygons, state whether it is (a)an equilateral polygon; (b) an equiangular polygon; (c)a regular polygon. B C A 6.2Plane Figures + Key Concept 6.2.8Key Concept 6.2.8

Index 1A_Ch6(31) 1.A solid is an object that occupies space. 2.The surfaces of a solid are called faces. 3.The line segment on a solid that is formed by any two intersecting faces is called an edge. 4.A point that is formed by 3 or more intersecting faces on a solid is called a vertex. A) Introduction edge face vertex 6.3Three-dimensional Figures + Index 6.3Index 6.3

Index 1A_Ch6(32) 1.We can use solid and dotted lines to draw rough 2-D figures of solids on a plane. B) Sketch the Two-dimensional (2-D) Representation of Simple Solids 2.We can also use isometric drawings to draw more accurate 2-D figures of solids on a plane. Isometric dotted paper Isometric grid paper 6.3Three-dimensional Figures + ExampleExample

Index 1A_Ch6(33) B) The face obtained by cutting a solid along a certain plane is called a cross-section of the solid. If we cut the solid at different positions, we may obtain different cross-sections. Note : If we obtain the same cross-sections by cutting a solid along certain direction, then the cross-sections are called uniform cross-sections. Different Cross-sections 6.3Three-dimensional Figures + ExampleExample + Index 6.3Index 6.3 Sketch the Two-dimensional (2-D) Representation of Simple Solids

Index 1A_Ch6(34) Use an isometric dotted paper to draw the 2-D representation of the box. 8 cm 4 cm 2 cm 6.3Three-dimensional Figures

Index 1A_Ch6(35) Use an isometric grid paper to draw the 2-D representation of the given solid. 6 cm 4 cm 2 cm 6.3Three-dimensional Figures + Key Concept 6.3.2Key Concept 6.3.2

Index 1A_Ch6(36) Which of the following faces represents the cross-section of the given solid when it is cut vertically along the blue line? ABC The cross-section is B. 6.3Three-dimensional Figures

Index 1A_Ch6(37) Draw the cross-section of the given solid when it is cut horizontally along the yellow line. Fulfill Exercise Objective  Draw the cross-section of a simple solid. The cross-section is : 6.3Three-dimensional Figures + Key Concept 6.3.3Key Concept 6.3.3

Index 1A_Ch6(38) A) Introduction to Polyhedra If all the faces of a solid are polygons, then that solid is called a polyhedron. Note : The polyhedra can be named by their numbers of faces. 6.4Polyhedra + ExampleExample + Index 6.4Index 6.4

Index 1A_Ch6(39) Determine which of the following solids is not a polyhedron. ABCD B 6.4Polyhedra + Key Concept 6.4.1Key Concept 6.4.1

Index 1A_Ch6(40) B) Making Models of Polyhedra We can use a net to make a model of polyhedron. (a)(b) For example, the net in Fig.(a) can be folded up to make a model of the polyhedron in Fig.(b). 6.4Polyhedra + ExampleExample + Index 6.4Index 6.4

Index 1A_Ch6(41) The diagram on the right is a polyhedron. Which of the following net do you think can make that polyhedron? A AB C 6.4Polyhedra + Key Concept 6.4.2Key Concept 6.4.2