Symmetry And Shapes

Introduction to Lines & Angles

• A **line** is straight, one-dimensional, and extends forever; a **line segment** has a start and end point.
• **Parallel lines** never intersect and are marked with arrows; **perpendicular lines** intersect at right angles (90°).
• **Acute angle**: less than 90°; **right angle**: exactly 90°; **obtuse angle**: greater than 90° but less than 180°.
• **Straight angle**: exactly 180°; **reflex angle**: greater than 180° but less than 360°.
• Angles at a point sum to 360°; angles on a straight line sum to 180°; angles that sum to 180° are **supplementary**.

Rotational Symmetry

• **Rotational symmetry** is the number of times a shape looks the same during a full 360° rotation; this number is the **order of rotational symmetry**.
• Use tracing paper with an arrow to track orientation; returning to the start counts as 1, so order is at least 1.
• A shape with order 1 has no rotational symmetry (only looks the same at the start).
• A regular octagon has rotational symmetry of order 8; a regular pentagon has order 5.

Lines of Symmetry

• A **line of symmetry** divides a shape into two mirror-image halves; folding along it makes the halves coincide.
• Some shapes have multiple lines of symmetry (e.g., square has 4, rectangle has 2).
• To complete a shape given a line of symmetry, reflect the given part across the line.
• Diagonal lines of symmetry require careful reflection; tracing paper can help.

Properties of Polygons

• Polygons are named by number of sides: triangle (3), quadrilateral (4), pentagon (5), hexagon (6), heptagon (7), octagon (8), nonagon (9), decagon (10).
• A **regular polygon** has all sides equal and all angles equal.
• Triangles: **equilateral** (3 equal sides/angles), **isosceles** (2 equal sides/angles), **right-angled** (one 90° angle), **scalene** (no equal sides).
• Quadrilaterals: **square** (4 equal sides, 4 right angles, 4 lines of symmetry, order 4 rotational symmetry).
• **Rectangle**: opposite sides equal, 4 right angles, 2 lines of symmetry, order 2 rotational symmetry.
• **Parallelogram**: opposite sides parallel and equal, opposite angles equal, no lines of symmetry, order 2 rotational symmetry.
• **Rhombus**: all sides equal, opposite sides parallel, opposite angles equal, 2 lines of symmetry, order 2 rotational symmetry.
• **Trapezium**: one pair of parallel sides; **isosceles trapezium**: non-parallel sides equal, 1 line of symmetry.
• **Kite**: two pairs of equal adjacent sides, one pair of equal opposite angles, 1 line of symmetry, order 1 rotational symmetry.

Properties of Circles

• Key terms: **circumference** (perimeter), **diameter** (line through centre, $2\times radius)$, **radius** (centre to circumference).
• **Chord**: line joining two points on circumference; **arc**: part of circumference; **sector**: region between two radii and an arc.
• **Segment**: region between a chord and an arc; **tangent**: line touching circle at one point only.
• Diameter $= 2 \times$ radius; ratio $circumference/diameter = \pi$.

Properties of 3D Shapes

• A **prism** has the same cross-section throughout; examples: cube (square cross-section), cuboid (rectangle), triangular prism, cylinder (circle).
• A **pyramid** has a flat base and sloping sides meeting at an apex; square-based pyramid, tetrahedron (triangular-based pyramid), cone (circular base).
• **Faces**: flat surfaces; **vertices**: corners; **edges**: lines where faces meet.
• Cube: 6 square faces, 12 edges, 8 vertices. Cuboid: 6 rectangular faces, 12 edges, 8 vertices.
• Triangular prism: 2 triangular + 3 rectangular faces, 9 edges, 6 vertices. Square-based pyramid: 1 square + 4 triangular faces, 8 edges, 5 vertices.
• Cylinder: 2 circular + 1 curved face, 2 edges, 0 vertices. Sphere: 1 curved face, 0 edges, 0 vertices.

Nets of Solids

• A **net** is a 2D shape that can be folded to form a 3D solid; the area of the net equals the surface area of the solid.
• Net of a cube: 6 squares; there are 11 possible nets, the most common is a cross shape.
• Net of a cuboid: 6 rectangles in three pairs (opposite faces equal); 54 possible nets.
• Net of a cylinder: two circles (top and bottom) and a rectangle (curved surface); rectangle length = circumference of circle.
• Net of a square-based pyramid: one square and four congruent triangles.

Types of Angles

Lines of Symmetry of a Square

Parts of a Circle

Net of a Cuboid