Points and Lines

Point

• A location in any space is represented by a dot

• It does not have any length, height, shape, or size. 

• Usually labeled with capital letters.

Line

• A series of points connected by a straight path. 

• A line is usually defined by two points and can be marked with a single lowercase letter or two capital letters. 

• A line has no thickness, and its length is undefined, meaning it can have an infinite number of points. 

Types of Points and Lines

There are different types of points in geometry. 

Types of Points

Collinear Points

• Three or more points that lie on a single straight line

Non-Collinear Points

• A group of points that do not lie on the same line 

Coplanar Points 

• A group of points that lie on the same plane

Non-Coplanar Points

• A group of points that do not lie on the same plane are non-coplanar points.

Types of Lines

There are different types of lines that can be distinguished easily based on their unique properties.

Horizontal Line 

• A horizontal line is a line that is mapped from left to right or right to left and parallel to the x-axis in a plane.

Vertical Line 

• A line that is mapped from up to down or down to up and is parallel to the y-axis in a plane is called a vertical line.

Intersecting Lines

• When two lines cross each other and meet at a point, they are known as intersecting lines. The point at which they meet is known as the point of intersection.

Perpendicular Lines

• When two lines intersect exactly at 90°, they are known as perpendicular lines.

Parallel Lines

• Two lines are said to be parallel if they do not intersect at any point and are equidistant.

Polygons 

A polygon is a 2D shape with straight sides. Some polygons have special names, for example, triangles and quadrilaterals.

Types of Polygons

Polygons can be regular or irregular. 

• Regular polygon = equal angles and sides

• Irregular polygon ≠  equal angles and sides

Common Polygons

• Triangle: 3 sides

• Quadrilateral: 4 sides (includes squares, rectangles, trapezoids, and parallelograms)

• Pentagon: 5 sides

• Hexagon: 6 sides

• Heptagon: 7 sides

• Octagon: 8 sides
  

Angles in Polygons 

Polygons can have both interior angles and exterior angles.

Interior Angles

An interior angle is the angle formed between two adjacent sides of a polygon on the inside of the shape. Each vertex (corner) of a polygon has an interior angle.

Finding the Sum of Interior Angles 

To find the sum of interior angles in a polygon, divide the polygon into triangles. The sum of interior angles in a triangle is 180°.

To find the sum of the interior angles of a polygon, multiply the number of triangles in the polygon by 180°.

Example

Calculate the sum of interior angles in a pentagon.

A pentagon has 5 sides, so it contains 3 triangles. 

The sum of the interior angles is: 180º × 3 = 540∘

The formula for calculating the sum of interior angles is:

(𝑛 − 2) × 180º  (where 𝑛  is the number of sides)

Exterior Angles 

If the side of a polygon is extended, the angle formed outside the polygon is the exterior angle. The sum of the exterior angles of a polygon is 360°.

Calculating the exterior angles of regular polygons

The formula for calculating the size of an exterior angle is 360 divided by the number of sides 

Example

This pentagon has 5 sides. To calculate the exterior angles, divide 360º by the number of sides

360º / 5 = 72º 

Triangles 

Triangles are 2D shapes with three sides. There are four different triangles with different properties.

Types of triangles 

Scalene 

• 3 sides of different lengths 

• 3 unequal angles.

Isosceles 

• 2 sides of equal length

• The dashes on the lines show they are equal in length

• The angles at the base of the equal sides are equal.

Equilateral 

• 3 sides of equal length. 

• The dashes on the lines show they are equal in length

• All of the angles are equal.

Right Angle

• A triangle that has a right angle

Labeling angles and sides

Letters can be used to label angles.

Example 

• AB and AC are line segments, and they meet at point A. 

Line segment: a line with 2 endpoints

• AB joins the points A and B.

• The angle between AB and AC is labelled BAC.

• The angle can written as BAC or BÂC or ∠BAC.

Interior & Exterior Angles

The angles inside a shape are called interior angles.

If the side of a triangle is extended, the angle formed outside the triangle is the exterior angle.

g = interior angle

h = exterior angle

g + h = 180º

The interior angle and its corresponding exterior angle always add up to 180°.

Sum of Interior Angles 

The sum of interior angles in a triangle is 180°.

To prove 𝑎 + 𝑏 + 𝑐 = 180º, draw a line parallel to one side of the triangle.

𝑑 = 𝑏  (alternate angles are equal)

𝑒 = 𝑐  (alternate angles are equal)

𝑎 + 𝑑 + 𝑒 = 180º (angles on a straight line add up to 180°)

So 𝑎 + 𝑏 + 𝑐 = 180º

These facts can be used to calculate angles.

Congruence 

Triangles are congruent if they are identical in size and shape. This means all corresponding sides and angles are equal.

Conditions for Congruence:

RHS: RIGHT ANGLE, HYPOTENUSE, SIDE

• Both have right angles

• Both have the same hypotenuse length

• Both have one other pair of sides which are equal 

ASA: ANGLE, SIDE, ANGLE

• 2 pairs of angles are equal 

• The side between the angles are the same 

SAS: SIDE, ANGLE, SIDE 

• 2 pairs of sides are equal 

• The angle between the pair of equal sides is the same
  

SSS: SIDE, SIDE, SIDE 

• All three pairs of sides are equal 
  
  

Pythagoras Theorem 

• Applies to right-angled triangles 

• States that the square of the length of the hypotenuse (c) is equal to the sum of the squares of the lengths of the other two sides (a and b).

  • a, b, and c are the lengths of the three sides

  • a² + b² = c²

Example

Given: A right-angled triangle with legs a = 3 cm and b = 4 cm.

Find: The hypotenuse c

• c² = a² + b² 

  • = 3² + 4² 

  • = 9 + 16 = 25

  • c² = 25 cm

    • Square root both sides = 25 → 5 cm

Circle Theorems

Circle theorems are properties that show relationships between angles within the geometry of a circle. 

Circle Theorem 1: The Alternate Segment 

The angle that lies between a tangent and a chord is equal to the angle subtended by the same chord in the alternate segment.

How to use it:

1. Locate the key parts of the circle for the theorem 

2. Use other angle facts to determine one of the two angles

3. Use the alternate segment theorem to state the other   missing angle 

Circle Theorem 2: Angles at the Center and at the Circumference

The angle at the centre is twice the angle at the circumference.

How to use the fact that the angle at the centre is twice the angle at the circumference:

1. Locate the key parents of the circle for the theorem

2. Determine the angle at the centre or the angle at the circumference

3. Use the angle at the centre to state the other missing angle 

Circle Theorem 3: Angles in the Same Segment

Angles in the same segment are equal.

How to use the fact that angles in the same segment are equal

1. Locate the key parts of the circle for the theorem.

2. Use other angle facts to determine an angle at the circumference in the same segment.

3. Use the angle in the same segment theorem to state the other missing angle.

Circle Theorem 4: Angles in a Semicircle

The angle in a semicircle is 90 degrees 

How to use the fact that angles in a semicircle equal 90º

1. Locate the key parts of the circle for the theorem.

2. Use other angle facts to determine angles within the triangle.

3. Use the angles in a semicircle theorem to state the other missing angle.
   

Circle Theorem 5: Chord of a Circle 

The perpendicular from the centre of a circle to a chord bisects the chord (splits the chord into two equal parts).

How to find missing lengths using chords

1. Locate the key parts of the circle for an appropriate circle theorem.

2. Use other angle facts to determine any missing angles.

3. Use Pythagoras’ theorem or trigonometry to find the missing length.

Circle Theorem 6: Tangent of a Circle

The angle between a tangent and radius is 90 degrees. Tangents which meet at the same point are equal in length.

How to use the tangent of a circle theorems:

1. Locate the key parts of the circle for the theorem.

2. Use other angle facts to determine the remaining angle(s) made with the tangent.

3. Use the tangent theorem to state the other missing angle.

Circle Theorem 7: Cyclic Quadrilateral

The opposite angles in a cyclic quadrilateral total 180º

How to use the cyclic quadrilateral theorem

1. Locate the key parts of the circle for the theorem.

2. Use other angle facts to determine one of the two opposing angles in the quadrilateral.

3. Use the cyclic quadrilateral theorem to state the other missing angle.