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Geometry

MAT1501 - Fundamental Mathematics · Geometry and the Measurement of Areas and Volumes

Geometry

Geometry is a branch of mathematics that deals with the study of shapes, sizes, and the properties of space. It is essential for understanding many concepts in fields such as engineering, architecture, and various sciences. This topic covers the basic elements of geometry, including lines, angles, and polygons, which are foundational for solving more complex problems in geometry.

Key idea: After studying this unit, you should be able to:

  • Define and identify lines, line segments, rays, and angles.
  • Classify angles based on their measures.
  • Recognise and create patterns using symmetry.
  • Understand the properties of polygons and their classifications.
  • Apply the concepts of congruence and similarity in polygons.

1.1 Lines and Angles

Understanding Lines

A line is a straight path that extends indefinitely in both directions and has no width. It is made up of an infinite number of points. A line segment is a part of a line that has two endpoints, while a ray is a part of a line that has one endpoint and extends indefinitely in one direction.

Example 1.1

Consider points A and B. The line segment AB consists of all the points between A and B, including A and B themselves. We write this as AB.

Line segment: AB

Angles

An angle is formed by two rays that share a common endpoint, known as the vertex of the angle. The amount of rotation from one ray to the other is the measure of the angle, typically expressed in degrees.

Example 1.2

If ray OA is rotated to ray OB, the angle formed is denoted as ∠AOB. If ray OA is rotated 90 degrees to reach ray OB, we say that ∠AOB is a right angle.

1.2 Classifying Angles

Angles can be classified based on their measures:

  • Acute Angle: An angle less than 90 degrees.
  • Right Angle: An angle equal to 90 degrees.
  • Obtuse Angle: An angle greater than 90 degrees but less than 180 degrees.
  • Straight Angle: An angle equal to 180 degrees.
  • Reflex Angle: An angle greater than 180 degrees but less than 360 degrees.

Example 1.3

If angle AOB measures 45 degrees, it is classified as an acute angle.

1.3 Symmetry

Symmetry is a property where an object is invariant under certain transformations, such as reflection or rotation. There are two main types of symmetry:

  • Reflection Symmetry: An object has reflection symmetry if it can be divided into two identical halves by a line (axis of symmetry).
  • Rotational Symmetry: An object has rotational symmetry if it looks the same after a certain amount of rotation about a central point.

Example 1.4

A butterfly has reflection symmetry; if you fold it along its centre line, both halves match perfectly.

Example 1.5

A square has rotational symmetry; it can be rotated by 90 degrees, 180 degrees, or 270 degrees and still look the same.

Watch out: Be careful not to confuse reflection symmetry with rotational symmetry. An object can have one or both types of symmetry.

1.4 Polygons

A polygon is a closed figure formed by connecting line segments at their endpoints. The segments are called sides, and the points where the sides meet are called vertices. Polygons are classified based on the number of sides they have:

  • 3 sides: Triangle
  • 4 sides: Quadrilateral
  • 5 sides: Pentagon
  • 6 sides: Hexagon
  • 8 sides: Octagon

Example 1.6

A triangle has three sides and three angles. If all sides are equal, it is an equilateral triangle.

1.5 Congruence and Similarity

Polygons are said to be congruent if they have the same shape and size, meaning all corresponding sides and angles are equal. They are similar if they have the same shape but not necessarily the same size, meaning corresponding angles are equal and corresponding sides are in proportion.

Example 1.7

If triangle ABC is congruent to triangle DEF, then all corresponding sides and angles are equal: AB = DE, BC = EF, AC = DF, and ∠A = ∠D, ∠B = ∠E, ∠C = ∠F.

Example 1.8

If triangle GHI is similar to triangle JKL, then the angles are equal: ∠G = ∠J, ∠H = ∠K, ∠I = ∠L, and the sides are in proportion: GH / JK = HI / KL = GI / JL.

Watch out: When determining similarity, ensure that you check both angles and sides. Just having equal angles does not guarantee similarity unless the sides are in proportion.

Summary

  • Geometry studies shapes and their properties.
  • Lines, line segments, and rays are fundamental concepts.
  • Angles are classified based on their measures.
  • Symmetry can be reflection or rotational.
  • Polygons are classified by the number of sides.
  • Congruent polygons have the same shape and size; similar polygons have the same shape but different sizes.

Check your understanding

  1. Define a line, line segment, and ray.
  2. Classify the following angles: 45 degrees, 180 degrees, and 95 degrees.
  3. What type of symmetry does a circle have?
  4. How do you determine if two polygons are similar?