Euclid is known as the "Father of Geometry" and laid the foundation for the study of mathematics. Among the various branches of geometry, Euclid's treatment of lines and angles remains a cornerstone in mathematical education. The fundamental concepts introduced by Euclid find practical use in various real-world scenarios which range from architecture and engineering to technology and art. In this chapter, we will delve into the fascinating world of Euclid's geometry, exploring the principles governing lines and angles and their enduring impact on the development of mathematics.
The word 'geometry' comes from the Greek word 'geo' meaning the 'earth' and 'metron' meaning 'to measure'. Geometry appears to have originated from the need for measuring lands. Our ancient civilizations studied geometry in various forms.
Euclid introduced the method of using deductive logical reasoning to prove mathematical results. The geometry of plane figures is known as Euclidean geometry.
Non-Euclidean geometry is not the same as Euclidean geometry. An example of non-Euclidean geometry is spherical geometry where geometry doesn't have straight lines.
Some important elements to remember:
Self-evident true statements used throughout mathematics and not specifically linked to geometry are called axioms while those specific to geometry are known as postulates.
Some of Euclid's axioms are as follows:
1) Equality Principle: If two entities are each equal to a third entity, then they are equal to each other.
Example: If A = C and B = C, then A = B.
Thus, here A, B and C are the same kind of entities.
2) Addition Principle: If you add equal entities to other equal entities, the sums are equal.
Example: If A = B and C = D, then A + C = B + D.
Thus, here A, B, C and D are entities.
3) Subtraction Principle: If you subtract equal entities from other equal entities, the remainders are equal.
Example: If A = B and C = D, then A − C = B − D.
4) Doubling Principle: If two entities are each double of the same entities, then they are equal to each other.
Example: If A = 2C and B = 2C, then A = B.
5) Halving Principle: If two entities are each half of the same entities, then they are equal to each other.
Example: If A = C2 and B = C2, then A = B.
6) Whole is Greater Principle: The whole entity is greater than any of its parts. If A is greater than B, then there exists entity C such that A is equal to the sum of B and C.
Example: If A > B, then there exists entity C such that A = B + C.
7) Coincidence Principle: Entities that exactly overlap or coincide with each other are equal.
Example: If two triangles overlap perfectly that means all their sides and angles match up exactly, then according to Euclid's Coincidence Principle, those two triangles are considered congruent.
Euclid's five postulates are as follows:
Postulate 1: Any Two Points
Any two points can be connected by a straight line. You can draw a straight line from one point to another.
Postulate 2: Extending a Line
A line with an endpoint can go on forever. If you have a line segment, you can keep extending it forever to make a longer line.
Postulate 3: Creating Circles
You can draw a circle with any centre (O) and radius (r).
Postulate 4: Equal Right Angles
All right angles are always equal to each other.
Postulate 5: Angle Sum for Lines
If a line intersects two other lines and the angles on one side add up to less than a right angle, those lines will eventually meet on that side.
If a straight line (PQ) intersects two other straight lines (l and m) in such a way that the sum of the interior angles (∠P and ∠Q) on one side is less than two right angles (180°), then those two lines will eventually meet if extended indefinitely. The intersecting lines (l and m) form an angle ∠A.
Note: Any two straight lines are either intersecting or parallel. If they cross, they can do so at any angle or they might meet at a right angle (perpendicular).
Concurrent Lines: Concurrent lines are two or more lines that come together at one common point on a flat surface in a plane. The point where these lines meet is called the point of concurrency.
Axioms of points and lines are as follows:
Axiom 1: Infinite Points
A line has infinite (countless) points along it.
Axiom 2: Infinite Lines from a Point
From any single point, you can draw infinite lines.
Axiom 3: One Line between Two Points
If you have two different points, there is only one line that goes through both of them.
If a line ‘l’ and a point ‘P’ that is not on that line, there is only one other line m that can pass through point P and never intersect with line l, then the lines l and m are parallel to each other.
An alternative statement for the mentioned axiom is expressed as
1) Two different lines (l and m) that intersect each other cannot both be parallel to a third line (n).
2) If three points A, B and C are in a straight line and C is between A and B, then the sum of AC and BC is equal to the total length AB. These three points A, B and C are collinear points.
If a point C lies between two points A and B, then AC + BC = AB.
3) If a point C is exactly in the middle of a line AB, then the distance from A to C is the same as the distance from C to B and both are half of the total length AB.
If C is the midpoint of a line segment AB, then AC = BC = 1/2 AB
Complementary and Supplementary Angles: Two angles are referred to as complementary angles when the total of their measures equals 90°.
Two angles are referred to as supplementary angles when the total of their measures equals 180°.
Adjacent Angles and Linear Pairs: When two lines meet and create a pair of angles next to each other, these angles are called adjacent angles. These adjacent angles are always supplementary and form a linear pair. Linear pairs are the adjacent angles formed by the intersection of two lines. The sum of angles of linear pairs is always 180°.
A transversal is a line that cuts across two or more lines and creates distinct points where the lines intersect.
If two straight lines don't intersect each other (parallel lines), then the angles formed by a third line that intersects them are related in a specific way. Euclid explained how these angles work when lines are parallel.
When a transversal intersects two parallel lines, the following angles are formed:
Corresponding Angles: The angles on the same side of the transversal and the same side of the parallel lines are equal.
∠a = ∠p, ∠b = ∠q, ∠c = ∠r and ∠d = ∠s are corresponding angles.
Vertically Opposite Angles: Angles that are opposite when two lines intersect are equal.
∠a = ∠c, ∠b = ∠d, ∠p = ∠r and ∠q = ∠s are vertically opposite angles.
Alternate Interior Angles: The angles on the inner sides of the lines but on opposite sides of the transversal are equal.
∠c = ∠p and ∠d = ∠q are alternate interior angles.
Alternate Exterior Angles: The angles on the outer sides of the lines but on opposite sides of the transversal are equal.
∠a = ∠r and ∠b = ∠s are alternate exterior angles.
Co-Interior Angles (Interior Angles on the Same Side): The angles on the same side of the transversal and inside the parallel lines add up to 180°. They are supplementary.
If ∠c and ∠q are co-interior angles, then ∠c + ∠q = 180°.
If ∠d and ∠p are co-interior angles, then ∠d + ∠p = 180°.
Axioms of angles are as follows:
1) Angle Based on Rotation: As an object rotates, it traces a circular path and angles are used to measure the amount of rotation. A half rotation around a semi-circle corresponds to 180° or π radians and a full rotation around a circle corresponds to 360° or 2π radians.
Based on the direction of measurement or the direction of rotation, angles can be of two types:
→ Positive Angles: An angle measured in the counterclockwise (anti-clockwise) direction is a positive angle.
→ Negative Angles: An angle measured in the clockwise direction is a negative angle.
2) Right Angles are Equal: If a square corner is like the corner of a book, all those corners are the same size. Euclid said that all these square corners or right angles are equal.
Theorem 1: The sum of all the angles formed on the same side of a line at a given point on the line is 180°.
Theorem 2: The sum of all the angles around a point is 360°.
Theorem 3: If two lines intersect each other, then the vertically opposite angles are equal.
Theorem 4: Lines which are parallel to the same line are parallel to each other.
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