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2.(v)  Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?  v) square 

v) Square:- A square is a quadrilateral in which all the four sides are equal and each internal angle is a right angle. To define the square, we must know about quadrilateral.  

2.(iv) Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? iv) radius of circle 

iv)  Radius of the circle : - The distance between the centre of the circle and any point on the circumference of the circle is called the radius of a circle 

2.(iii)  Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them? iii) line segment

Yes, there are other terms that are needed to be defined first which are: Plane: A plane is a flat surface on which geometric figures are drawn. Point: A point is a dimensionless dot which is drawn on a plane surface. Line: A line is collection of n number of points which can extend in both the directions and has only one dimension. iii)  line segment : - A straight line with two end points...

2.(ii) Give a definition for each of the following terms. Are there other terms that need to be defined first? What are  they, and how might you define them? (ii) perpendicular lines 

Yes, there are other terms that are needed to be defined first which are: Plane: A plane is a flat surface on which geometric figures are drawn. Point: A point is a dimensionless dot which is drawn on a plane surface. Line: A line is the collection of n number of points which can extend in both the directions and has only one dimension. ii) perpendicular line:- If two lines intersect with each...

Q2   Does Euclid’s fifth postulate imply the existence of parallel lines? Explain.

According to Euclid's 5 postulates, the line PQ  falls on lines AB and CD such that the sum of the interior angles 1 and 2 is less than 180° on the left side of PQ. Therefore, the lines AB and CD will eventually intersect on the left side of PQ Now, If  then, the line never intersects with each other. Therefore, we can say  that lines AB and CD are parallel .to each other 

1. How would you rewrite Euclid’s fifth postulate so that it would be easier to understand?

Euclid's postulate 5: If a straight line falling on two straight lines makes the interior angles on the same side of it taken together less than two right angles, then the two straight lines, if produced indefinitely, meet on that side on which the sum of angles is less than two right angles. Now, in an easy way Let the line PQ in falls on lines AB and CD such that the sum of the interior...

7.  Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’? (Note that the question is not about the fifth postulate.)

Axiom 5 states that the whole is greater than the part. Lets take A = x + y + z where A , x , y , z all are positive numbers Now, we can clearly see that A > x , A > y , A > z  Hence, by this we can say that the whole (A) is greater than the parts. (x , y , z)  

6.  In Fig. 5.10, if AC = BD, then prove that AB = CD.

      

From the figure given in the problem, We can say that AC = AB + BC  and  BD = BC + CD Now, It is given that AC = BD Therefore, AB + BC = BC + CD Now,  According to Euclid's axiom, when equals are subtracted from equals, the remainders are also equal. Subtracting BC from both sides. We will get  AB + BC - BC = BC + CD - BC AB = CD Hence proved 

5. In Question 4, point C is called a mid-point of line segment AB. Prove that every line  segment has one and only one mid-point.

Let's assume that there are two midpoints  C and D  Now, If C is the midpoint then,  AC = BC And In the figure given above, AB coincides with AC + BC. Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB From this, we can say that  2AC = AB                                     ...

4.  If a point C lies between two points A and B such that AC = BC, then prove that AC = 1/2  AB. Explain by drawing the figure.

It is given that AC = BC  Now, In the figure given above, AB coincides with AC + BC. Also, Euclid’s Axiom (4) says that things which coincide with one another are equal to one another. So, it can be deduced that AC + BC = AB Now, 2AC = AB                                         Therefore, Hence proved 

3.   Consider two ‘postulates’ given below:
      (i) Given any two distinct points A and B, there exists a third point C which is in  between A and B.
      (ii) There exist at least three points that are not on the same line.
      Do these postulates contain any undefined terms? Are these postulates consistent?
      Do they follow from Euclid’s postulates? Explain.

There are various undefined terms in the given postulates.: 1)  There is no information given about the plane whether the points are in the same plane or not. 2) There is the infinite number of points lie in a plane. But here the position of the point C has not specified whether it lies on the line segment joining AB or not. Yes, these postulates are consistent when we deal with these two...

2. Give a definition for each of the following terms. Are there other terms that need to be defined first? What are they, and how might you define them?
(i) parallel lines 

Yes, there are other terms that are needed to be defined first which are: Plane: A plane is a flat surface on which geometric figures are drawn. Point: A point is a dimensionless dot which is drawn on a plane surface. Line: A line is the collection of n number of points which can extend in both the directions and has only one dimension. i) Parallel line:- If the perpendicular distance between...

1.   Which of the following statements are true and which are false? Give reasons for your answers.
       (i) Only one line can pass through a single point.
       (ii) There are an infinite number of lines which pass through two distinct points.
      (iii) A terminated line can be produced indefinitely on both the sides.
      (iv) If two circles are equal, then their radii are equal.
      (v) In Fig. 5.9, if AB = PQ and PQ = XY, then AB = XY.

  

i) FALSE  Because there is the infinite number of lines that can be passed through a single point. As shown in the diagram below ii)  FALSE  Because only one line can pass through two distinct points. As shown in the diagram below iii)  TRUE Because a terminated line can be produced indefinitely on both sides. As shown in the diagram below iv) TRUE Because if two circles are equal,...
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