Chapter-5 Introduction to Euclid Geometry

Explanation: 

(i) Yes. 

 A point is a unique position in space, and a line is a series of infinite points extending infinitely in two opposite directions. Therefore, there can be only one straight line through a point. 

(ii) False. 

 Only one straight line can pass through two different points. However, there are an infinite number of points on this straight line. 

(iii) Yes. 

A completed line is a line segment that has a start point and an end point. You can create unlimited lines on either side by extending the line in either direction. 

(iv) Yes.  

If two circles have the same radius, they are congruent and have all corresponding parts equal, including the radii. 

(v) True.  

If AB = PQ and PQ = XY, then by transitive equality we can say that AB = XY. This property says that if a = b and b = c, then a = c. Therefore, AB and XY have the same length PQ and are therefore equal.

Explanation: 

Yes, these are the following terms which are required to be defined before one can understand the given ones. 

• Point: A point is a dimensionless geometric object with no length, width, or thickness. It is generally indicated by a dot or slash and is named with a letter or combination of letters. 

• Line: A line is a one-dimensional geometric object, meaning it has a length but no width or height, that further that it can only be a straight line. A straight line can be thought of as an infinite series of connected points extending endlessly in both directions.it is defined as the shortest path between two points. 

• Plane: A plane is a two-dimensional flat surface that extends endlessly in all directions. It can be defined as a set of points equidistant from two fixed points, or a plane can be defined as a set of points that form a plane when connected by lines. 

(i) Parallel lines: Parallel lines are lines in a plane that never intersect and are always a constant distance apart. An important property of parallel lines is that parallel lines can extend infinitely in either direction or never intersect. This property is called the parallel postulate, one of his five postulates of Euclidean geometry. 

(ii) Perpendicular lines: Perpendicular lines are lines that intersect at a 90-degree angle in the plane. Important properties of perpendicular lines are 

• They form four right angles at their intersection.  

• A line perpendicular to any line is also perpendicular to the other at the intersection. 

(iii) Line segment: A line segment is a part of a line segment that has two endpoints and has a measurable length. 

• Length: It has a measured length. 

• Endpoints: A line segment has two separate endpoints that mark the start and end points of the segment. 

• Direction: A line segment has a specific direction defined by its endpoints. 

• Crossroads: 

• A line segment can intersect other line segments, lines, or planes at its endpoints. 

(iv) Radius of circle: The radius of a circle is the distance between the center of the circle and any point on the circumference. It is represented by the letter "r". The radius of a circle is the distance from the center of the circle to any point on the circumference. It is a constant fixed length across the circle and is used to calculate the perimeter, area and other properties of the circle. 

(v) Square: A square is a quadrilateral with four equal sides and four internal angles that are all right angles (90 degrees). It has congruent sides and angles since it is a regular polygon. Four lines of symmetry, equal diagonals that cross at right angles, and an area that is equal to the square of the side length are all characteristics of a square.

Explanation: 

These postulates do indeed contain undefined terms.   

There are two of the following: 

• It is not mentioned where point C is, i.e., whether or not point C is on the line segment connecting points A and B. 

• About whether the points are in the same plane or not, nothing is known about it. 

• Yes, these postulates are consistent. 

• According to the first postulate, any two distinct points A and B can be connected at point C. It is possible to choose any point on a line segment between points A and B to serve as point C, hence this is always true. 

• According to the second postulate, there must be at least three points that do not lie along the same line. This is also valid since the postulate can be satisfied by any three non-collinear points in a plane. 

No, these axioms do not comply to Euclid's axioms. The five essential presumptions about geometry that make up Euclid's postulates were first outlined in his book "Elements." Although there are some similarities between these postulates and Euclid's postulates, they are not the same. Instead, one could think of these postulates as basic axioms of geometry.                                        

Explanation: 

Here, we have points A and B on a line, and point C is between them such that AC = BC. We need to prove that AC = ½ AB. 

Since AC = BC, we can add them together to get: 

AC + BC = 2AC 

We also know that AB = AC + CB, since AC and CB make up the entire distance from A to B. We can substitute AC + BC for both AC and CB in this equation: 

AB = AC + CB = AC + AC = 2AC 

Now we have AB = 2AC. To prove that AC = ½ AB, we just need to divide both sides of the equation by 2: 

AC = ½ AB 

Therefore, we have proved that if a point C lies between two points A and B such that AC = BC, then AC = ½ AB. 

Explanation: 

Proof by contradiction can be used to prove that every line segment has one and only one midpoint. 

Assume you have a line segment AB with two distinct midpoints, say C and D. 

By definition of midpoint, we have AC = CB and AD = DB. 

 Now, let's consider the triangle ACD. 

We know from triangle properties that sum of its two side is always greater than third side, therefore; 

AC + AD > CD 

Substituting AC = CB and AD = DB, we get: 

CB + DB > CD 

But by definition of midpoint, we have CD = AB/2. Therefore, we have: 

CB + DB > AB/2 

Similarly, we can consider the triangle BCD and obtain: 

BC + BD > AB/2 

Adding these two inequalities together, we get: 

2(CB + DB) > AB 

But this contradicts the fact that CB + DB = AB/2, which follows from the assumption that C and D are both mid-points of AB. Therefore, our initial assumption that AB has two distinct mid-points is false. 

Hence, we can conclude that every line segment has one and only one mid-point.

Explanation: 

We are given that AC = BD. We can see from the figure that AC + CD = AD and AB + BD = AD. 

We can substitute AC = BD in the first equation to get AC + AC = AD, which simplifies to 2AC = AD.  

Similarly, we can substitute AC = BD in the second equation to get AB + AB = AD, which simplifies to 2AB = AD. 

Therefore, we have 2AC = AD and 2AB = AD. Adding these two equations, we get: 

2AC + 2AB = 2AD 

Dividing both sides by 2, we get: 

AC + AB = AD 

But we also know that AC + CD = AD and AB + BD = AD. We can substitute AC = BD in the second equation to get AB + AC = AD. 

Therefore, we have AC + CD = AB + AC, which simplifies CD = AB. 

Hence, we have proved that if a line has 4 points A, B, C and D in it such that AC = BD, then AB = CD.  

Explanation: 

Axiom 5, "The whole is greater than the part," is regarded as a "universal truth". 

As it is a fundamental principle of mathematics that applies to all aspects of life. This axiom is not only used in geometry, but it is also used in economics, physics, and engineering.  

These are two examples for understanding it clearly; 

• if you cut a pizza into slices, the total area of the slices will always be less than the total area of the pizza.  

• Similarly, if you have a bag of candies and take some of them out, the remaining candies in the bag will always be greater than the candies you took out.

Explanation: 

The fifth postulate of Euclid deals with parallel lines. According to this rule, if a straight line intersects another straight line and the interior angles on the same side are less than two right angles when added together, the two straight lines will continue to intersect on that side indefinitely. 

Because they are lines that never cross one other, regardless of how far they are extended, parallel lines are significant in geometry. 

• They can consist of two or more lines and are always spaced apart uniformly. 

• We can draw a line through a point X that is parallel to line A if it does not lie on line A.  

• actually, only one line that passes through point X and is perpendicular to line A is possible.

Explanation: 

Yes, Euclid’s fifth postulate does imply the existence of parallel lines. 

If the sum of the interior angles is equal to the sum of the right angles, then the two lines will not meet each other at any given point, hence making them parallel to each other. 

∠1+∠3 = 180o 

Or ∠3+∠4 = 180o