Geometry

GEOMETRY ​

By : kasskassmaths​

​General Mathematics Syllabus

Section : #​8, Objectives 1, 4 - 6

Table of Contents

Concepts relating to geometry - Slide ​3

​Angles - Slide 7

​Parts of a solid/shape - Slide 12

Quadrilaterals - Slide​ 14

Triangles ​- Slide​ 15

Symmetry - Slide​ 21​

Properties of a Circle - Slide ​26

​Circle Theorem - Slide 28

Tangent Theorem - Slide ​38

Past Paper Questions to Practice - Slide​ 41

Concepts related to geometry

(1) Points : A point is a location in space or on a surface. A point is said to have position but no size. It is often described by its co-ordinates (x,y), where x is the point's position along the x axis and y is the point's position along the y axis. ​

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(2) Line Segment :

This is a part of a straight line between 2 given points. ​

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(3) Line : A line is said to have length but no breadth or thickness. A line spreads indefinitely in both directions.

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(4) Parallel Lines : These are lines in a plain that do not meet. The lines do not intersect or touch each other at any point.​

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​(5) Intersecting Lines :

These are 2 lines that share exactly one point. This point is called the point of intersection.

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(6)Perpendicular Lines :

These are lines that are at right angles to each other.

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(7) Ray :

A ray is a straight line extending from a point called the origin.

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(8) Curves :

This is defined as a line that is not straight.

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(9) Planes :

This is a flat 2 - dimensional surface that extends infinitely. ​

ANGLES : Types of Angles

Angles associated with parallel lines.

(1)Corresponding angles - angles that are in corresponding positions when a transversal cuts two parallel lines. The angles in corresponding positions are equal.

(2)Alternate angles - angles enclosed by a Z when a transversal cuts two parallel lines.

(3)Co - interior angles - ​occur in between 2 parallel lines when they are intersected by a transversal. The 2 angles that occur on the same side of the transversal always add up to 180.

If two parallel lines are cut by a transversal as shown in the diagram, we refer as follows to the angles formed:

• z and x (or u and v) are alternate angles

• x and y are corresponding angles

• u and x (or z and v) are co-interior angles

• y and z are vertically opposite angles

Properties of Angles and straight lines

1) When 2 straight lines intersect the opposite angles are equal. The angles A and C are vertically opposite angles. Similarly the angles B and D are also vertically opposite angles.

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2) When 2 parallel lines are cut by a transversal :

(a) the corresponding angles are equal

a = l, b = m , c = p, d = p.

(b)​ the alternate angles are equal

d = m, c = l

(c) the interior angles are supplementary ​

​​d + l = 180, c + m = 180

Parts of a Shape / Solid

Face - any flat surface(s) on a solid figure

Edge ​- the line segment(s) where 2 faces meet

Vertex - a point where several plates meet in a point​

​Name	Edges​	Vertices​	Faces​
​square-based pyramid	​8	​5	​1 square 4 triangles
triangular- based pyramid​	6​	4​	4​ triangles
triangular prism​	9​	​6	2 triangles 3 rectangles

Quadrilaterals

​Quadrilateral	​Properties
​Square	opposite sides are parallel, 4 sides are equal, 4 right angles
​Rectangle	​​4 right angles and opposite sides are equal and parallel
​Parallelogram	​opposite sides are equal and parallel, opposite angles are equal
Trapezium​	one pair of opposite sides parallel
Rhombus​	​opposite sides are parallel, 4 sides are equal, opposite angles are equal​
Kite​	2 pairs of equal adjacent sides, one pair of equal opposite angles​

​Triangles

Types of Triangles

Congruent Triangles

​Two triangles are said to be congruent if they are equal in every respect. There are 4 cases of congruency. Two given triangles can be determined congruent if they have at least one / out of these 4 characteristics :

(1) S.S.S. - ​the 3 sides of one triangle is equal to the corresponding 3 sides of the other triangle.

​(2) S.A.S. - two sides and the including angle of one triangle are equal to the corresponding two sides and included angle of the other triangle.

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(3)​ A.A.S. - two angles and a side of one triangle are equal to the corresponding two angles and side of the other triangle.

(4) R.H.S - both triangles are right-angled, and the hypotenuse and a side of one triangle is equal to the hypotenuse and corresponding side of the other triangle.​

Similar Triangles

E.GS

Find the side marked x.

Triangle ABC :

C = 180 - (50 + 70) = 60

Triangle DEF :

E = 180 - (50 + 60) = 70

Therefore triangle ABC and DEF are ​similar because their corresponding angles are the same. Hence the ratio of their sides are also the same.

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Symmetry

Line of Symmetry

A figure has line symmetry if it can be folded along a line and the two halves match. The folded line is the line of symmetry.​

Note :

A circle has an infinite numbe​r of lines of symmetry.

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Rotational Symmetry

A plane figure is said to have rotational symmetry of a certain order if the plain figure maps onto itself under rotations through stated angles about a common center. ​

ORDER 1

All plain figures have rotational symmetry of order 1. When rotated at 360 degrees it will map back onto itself.

E.g. If a rectangle ​rotates 360 degrees at point 'o' it will map onto itself. It is said to have rotation symmetry of order 1.

ORDER 2

If a shape maps onto itself when rotated 360 and 180 degrees it is said to have rotational symmetry of order 2. ​

E.g. a rectangle ​

​ORDER 3

A rotational symmetry of order 3 is noted when a shape is rotated about its origin through 3 angles. ​

E.g. An equilateral triangle ​

ORDER 4

This is when the shape maps unto itself on all 4 turns.​

​E.g. A square

Point symmetry

A plane figure is said to have this if the plane figure maps onto itself after a rotation through 180 degrees about a central point, therefore if a plane figure posses ​rotational symmetry of order 2, it has point symmetry as well.

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Properties of a Circle

​When a whole circle is cut in 2, it gives 2 semi-circles.

​Sectors are bounded by 2 radii and one arc.

​Equal or congruent Circles all have the same radii​.

Circle Theorem

Theorem 1

​The angle at the center of the circle is twice the angle at the circumference standing on the same arc.

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angle 2x = 2 multiply by angle x

or

angle x = angle 2x divided by 2

or

DBC = 2 DAB

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E.g.

Theorem 2

Angles sub-tended by the diameter (or semi-circle) = 90 degrees. The hypotenuse of the right angle triangle passes though the center of the circle and AC is a diameter of the circle. Hence the triangle is a right angle triangle.​

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ABC = 90 degrees

E.g.

Theorem 3

Inscribed angles subtended by the same arc are equal. That is, angles at the circumference of a circle standing on the same arc are equal.

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E.g.

Theorem 4

​The opposite angles of a quadrilateral are supplementary. (angles that sum to 180 degrees)

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angles A an D are opposite angles

Therefore A + D = 180 degrees

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angles B and E are opposite

Therefore B + E = 180 degrees

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E.g.

Theorem 5

​The exterior angles of a quadrilateral is equal to the opposite interior angle.

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E.g.

Tangent Theorem

Theorem 1

​The radius of a circle drawn to the point of contact to a tangent is perpendicular to the tangent.

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APB = tangent

OP = radius

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Theorem 2 ​

The lengths of 2 tangents from an external point to the points of contact on the circle are equal.

​x

​x

​y

​y

​tangent

​tangent

Theorem 3 - The angle between a tangent to circle and a chord at the point of contact is equal to the angle in the alternate segment of the circle.​

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​DCA = CBA

ECB = CAB​

Past Paper Questions To Practice

May/June 2024 No. 9(a)
May/June 2023 No. 9(a)
May / June 2022 No. 9(a)

July 2021​ No. 9(a)

​June 2019 No. 9(a)

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