Introduction to Euclid's Geometry Important Questions

Very Short Answer Questions [1 Mark Questions]

Ques. Which of these statements is not in line with Euclid’s axiom?

1. If equals are added to equals, the whole is equal.
2. If equals are subtracted from the equals, the remainder is always equal.
3. Things which are equal to one same thing are equal to one another.
4. The whole is lesser than the part.

Ans: Option d) is the correct answer.

Explanation: The statement is given in option d) says the whole is lesser than the part. But, the whole can never be lesser than the part. In fact, parts make the whole and if we divide the whole then it is parts. So, the whole is never less than the part. Hence, option d is the right answer. 

Ques. Which of the following statements are true?

1. An infinite number of lines can be there that pass through 2 distinct points.
2. Only 1 line can pass through a single point.
3. We can produce a terminated indefinitely on both sides
4. If two circles are equal, then their radii are always unequal.

Ans: Option c) is the correct answer.

Explanation: Every other statement except the statement given in option c is false. Because a single point can be crossed or passed by many lines. Similarly, when two circles are equal, then, their radii are also equal, instead of being unequal. But, a terminated line can be produced indefinitely on both sides. Hence, option c is the correct answer.

Ques. If a line is drawn from the center of the circle to any point on its circumference. Then, this line is called:

1. Radius
2. Diameter
3. Sector
4. Arc

Ans: Option a) is the correct answer.

Explanation: Radius, not diameter, sector, or arc, is the line that is drawn from the center of the circle to any point on its circumference. Hence, option a is the correct answer.

Ques. How many Euclid’s Postulates are there?

1. Three
2. Four
3. Five
4. Six

Ans: Option c) is the correct answer.

Explanation: A total of five postulates are there that are given by Euclid in the introduction to Euclid Geometry. Hence options c is the correct answer.

Ques. Euclid’s fifth postulate is

1. The whole is greater than part
2. All right angles are equal to one another.
3. If a straight line falling on two straight lines makes the interior angles on the same side. So, if taken together less than two 90 degree angles. Then, the two straight lines meet on that side on which the sum of angles is 90-degree angle.
4. A circle may be described with any center and any radius

Ans: Option c) is the correct answer.

Explanation: The fifth postulate of Euclid is that:

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. If produced indefinitely, the two straight lines meet on the side on which the sum of angles is less than and not more than two 90 degree angles i.e., right angles. Hence, option c is the correct answer.

Ques. The three steps from solids to points are: 

1. Lines – points – surfaces – solids
2. Lines – surfaces – points – solids
3. Solids – lines – surfaces – points 
4. Solids – surfaces – line point

Ans: Option d is the correct answer.

Explanation: The three steps from solids to points are as follows. Solids to surfaces and then, surfaces to line points. Because we have to move from solids to points and points are actually the end point. Hence, option d is the correct answer.

Ques. Axioms are assumed 

1. Theorems
2. Definitions
3. Universal truths specific to geometry
4. Universal truths in all branches of mathematics

Ans: Option d is the correct answer.

Explanation: All the Axioms are assumed as universal truths in all branches of mathematics. In other words, there isn't any need for mathematical deduction to prove them. Hence, Option d is the correct answer.

Ques. Solve the equation a – 15 = 25. Explain which axiom do you use here and how?

Ans:

a – 15 = 25

By adding 15 to both sides, we obtain

a – 15 + 15 = 25 + 15 [using Euclid’s second axiom]

The second axiom of Euclid: 

If equals are added to equals, the wholes are equal.

We added 15 on both sides, hence the answer is a = 40.

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Short Answer Questions [2 Marks Questions]

Ques. Ram and Ravi have the same weight. If both of them gain weight by 2 kg each, how will their new weights be compared?

Ans: Already given: Ram and Ravi have the same weight. 

Let y kg be the weight of each of Ram and Ravi.

After adding 2 kg,

The weight of Ram and Ravi will be 

(y + 2) kg each.

According to Euclid’s second axiom, if equals are added to equals, the whole is equal.

So, the weight of Ram and Ravi are again equal. We can compare it the same way we were comparing the weights earlier.

Ques. In the given figure, name the following:

1. Four collinear points
2. Five rays
3. Five line segments
4. Two pairs of non-intersecting line segments.

Ans: 

1. D, E, F, G, and H, I, J, and K are the four collinear points.
2. DG, EG, FG, HK, IK are five rays.
3. DH, EI, FJ; DG, HK are five line segments.
4. (DH, EI) and (DG, HK) is the two pairs of non-intersecting line segments.

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Long Answer Questions [3 Marks Questions]

Ques. Why is Axiom 5, in the list of Euclid’s axioms, considered a ‘universal truth’?

Ans. Axiom 5 is that the whole is greater than the part. Axiom 5 in the list of Euclid's axioms is considered a universal truth because it holds true throughout the field of mathematics. Moreover, it is proven true even in other disciplines such as natural sciences as well.

Ques. Rohan’s maid has two children of the same age. Both of them have an equal number of dresses. On his birthday, Rohan has a plan to give both of them the same number of dresses. How many dresses do you think each one of them will have after Rohan’s birthday? Which Euclid’s axiom did you used to answer this question? What value is depicted there by Rohan's behavior? Write one more Euclid’s axiom.

Ans. Now, a number of questions are asked to us. Let us first note down the questions asked there.

• The number of dresses each one of them will have after Rohan’s birthday? 
• Which Euclid’s axiom is used to answer this question? 
• What value is Rohan depicting by doing so?

Now, Rohan’s maid has two children of the same age group and both of them have an equal number of dresses. Rohan on his birthday plans to give both of them the same number of dresses.

Therefore, by using Euclid’s Axiom there, if equals are added to equals, then the whole is equal. Hence, again both of them have an equal number of dresses. 

Values: The Values depicted by Rohan are caring and other social values. 

3rd Axiom: According to Euclid’s Axiom 3, if equals are subtracted from equals, then the remainders are also equal.

Ques. Three lighthouse towers are located at points A, B, and C on the section of a national forest to protect animals from hunters in the forest department as shown in the figure. Which value is the department exhibiting by locating extra towers? How many straight lines can you draw from A to C? Which of the Euclid Axiom you used to state the required result. Give one more. Postulate.

Ans. As we know, a number of questions are asked to us. Let us first note down the questions asked there,

• Which value is the department exhibiting by locating extra towers?
• How many straight lines can be drawn from A to C?
• State the Euclid Axiom which states the required result. Give one more.

As we re-read and comprehend the question, we can understand that one and only one line can be drawn from A to C. According to Euclid’s Postulate, a straight line may be drawn from anyone point to any other point. 
One more postulate says a circle may be described with any center and any radius.” 
Value: Wildlife is a part of our environment and conservation of each of its elements is needed for ecological balance. Hence, the value of conservation of wildlife and the environment are the values the department is exhibiting by locating extra towers.

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Very Long Answer Questions [5 Marks Questions]

Ques. Which of the following statements are true and which are false? Give reasons for your answers.

1. Only one line can pass through a single point.

Ans: False

Explanation: Many, i.e., infinite lines can pass through a single point. And that is the correct statement. Hence, the statement that only one line can pass through a single point is false.

2. There are an infinite number of lines that pass through two distinct points.

Ans: False 

Explanation: Euclid's first postulate says that there is a unique line that passes through two distinct points. Hence, this statement says that there are an infinite number of lines that pass through two distinct points is false. Because it contradicts the first postulate of Euclid.

3. One can produce a terminated line on both sides till infinity. 

Ans: True

Explanation: Euclid's second Postulate says that a terminated line can be produced indefinitely. Hence, this statement saying a terminated line can be produced indefinitely on both sides is correct.

4. If two circles are equal, then their radii are equal.

Ans: True

Explanation: To rationalize this, we will first consider two circles with the same radii.
We can observe and analyze that, when we make the two circles overlap with each other, we will get a superimposed figure of the two circles. Hence, we can conclude that the radii of both the circles will also be equal as the two circles are equal.

5. if AB=PQ and PQ=XY, then AB=XY

Ans: True

Explanation: It is already given to us that AB=PQ and PQ=XY.
Now, we need to consider Euclid's axiom that, given two distinct points, there is a unique line that passes through them.
Hence, we can conclude that AB, PQ, and XY are the lines with the same dimensions. Thus, if AB=PQ and PQ=XY, then AB=XY.

Ques. Provide a definition for each of the following terms. Is there any other term/ terms that need to be defined first? What are the terms, and how might you define them?

1. Parallel lines

Ans: When the perpendicular distance between two or more lines is always constant, then, these two or more lines are said to be parallel. We can also say that the lines that never intersect with each other are known as parallel lines.
As we can see, we need to define the line first, in order to define parallel lines.

2. Perpendicular lines

Ans: Two lines are considered as perpendicular lines when the angle between these two lines is 90 or a right angle.
As we can observe, we would have to define line and angle first, in order to define perpendicular lines.

3. Line segment

Ans: A line of a fixed dimension between two given points is known as a line segment.
As we can observe, we would have to define line and point first, in order to define a line segment.

4. Radius of a circle

Ans: The radius of a circle is the distance of any point lying on the boundary of a circle from the center of the circle.
It is apparent that we will need to define the circle and center of a circle, in order to define the radius of a circle.

5. Square

Ans: A quadrilateral with all the four sides equal and all the four angles of 90 degrees is known as a square.
As we can observe, we will need to define quadrilateral and angle first, in order to define a square. 

Ques. What are the five postulates of Euclid’s Geometry?

Ans: The five postulates of Euclid's geometry are as follows:

1. Things that are equal to the same thing are also equal to one another.
2. If equals are added to equals, then the whole is also equal.
3. If equals are subtracted from equals, the remainder is also equal.
4. Things which coincide with one another are also equal to one another.
5. The whole is always greater than the part.