Class 9 Maths Chapter 5 Previous Year Questions - Introduction to Euclid`s Geometry

Sol: It states that: “If a straight line falling on two straight lines make 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 the side on which the sum of the angles is less than two right angles.”

Sol: The boundaries of a surface are curves.

Sol: The first axiom explains the above statement.

Sol: The option (ii) is the Euclid’s fifth postulate.

Sol: (i) The universal truths in all branches of Mathematics.

Sol: Pythagoras was the student of ‘Thales’.

Sol: A triangle.

Sol: An angle.

Sol:

We can prove it by Euclid’s axiom 3. “If equals are subtracted from equals, the remainders are equal.”

We have AD = BC

⇒ AD – CD = BC – CD

⇒ AC = BD

Sol:

True.

∵ By Euclid’s first axiom “Things which are equal to the same thing are equal to one another”.

∴ AB = PQ and XY = PQ ⇒ AB = XY

Sol:

∵ C is the mid-point of AB

AB = 2AC

Also, D is the mid-point of XY

XY = 2XD

By Euclid’s sixth axiom “Things which are double of same things are equal to one another.”

∴ AC = XD = 2AC = 2XD

⇒ AB = XY

Sol:
Here, ∠3 = ∠4, ∠1 = ∠3 and ∠2 = ∠4. Euclid’s first axiom says, the things which are equal to equal thing are equal to one another. So ∠1 = ∠2.,

Sol:
Given that AB = BC and BX = BY
By using Euclid’s axiom 3, equals subtracted from equals, then the remainders are equal, we have
AB – BX = BC – BY
AX = CY

Sol:
We have
AC = DC
CB = CE
By using Euclid’s axiom 2, if equals are added to equals, then wholes are equal.
⇒ AC + CB = DC + CE
⇒ AB = DE. 

Sol:

 In the given figure, AC coincides with AB + BC. Also, Euclid’s axiom 4, states that things which coincide with one another are equal to one another. So, it is evident that:
AB + BC = AC.

Sol:

Two lines are said to be parallel, when the perpendicular distance between these lines is always constant or we can say that the lines that never intersect each other are called as parallel lines.

We need to define line first, in order to define parallel lines.

Ans: Two lines are said to be perpendicular lines, when angle between these two lines is 90^∘.

We need to define line and angle, in order to define perpendicular lines.

Sol:

 A line of a fixed dimension between two given points is called as a line segment.

We need to define line and point, in order to define a line segment.

Sol:

The distance of any point lying on the boundary of a circle from the center of the circle is called as radius of a circle.

We need to define circle and center of a circle, in order to define radius of a circle.

Sol:

A quadrilateral with all four sides equal and all four angles of 90^∘ is called as a square.

We need to define quadrilateral and angle, in order to define a square.

Sol:

 We are given that a point C lies between two points B and C, such that AC = BC.
We need to prove that AC = 1/2AB⋅
Let us consider the given below figure.

We are given that AC = BC−…(i)
An axiom of the Euclid says that "If equals are added to equals, the wholes are equal."
Let us add AC to both sides of equation (i).
AC + AC = BC + AC.
An axiom of the Euclid says that "Things which coincide with one another are equal to one another." "
We can conclude that BC+AC coincide with AB, or
AB = BC + AC.…(ii)
An axiom of the Euclid says that "Things which are equal to the same thing are equal to one another."
From equations (i) and (ii), we can conclude that
AC + AC = AB, or 2AC = AB
An axiom of the Euclid says that "Things which are halves of the same things are equal to one another."
Therefore, we can conclude that AC = 1/2AB

Sol:

We are given that AC = BD
We need to prove that AB = CD in the figure given below.

From the figure, we can conclude that
AC = AB + BC, and BD = CD + BC.
An axiom of the Euclid says that "Things which are equal to the same thing are equal to one another."
AB + BC = CD + BC
An axiom of the Euclid says that "when equals are subtracted from equals, the remainders are also equal."
We need to subtract BC from equation(i) to get AB + BC − BC = CD + BC− BC

AB = CD

Therefore, we can conclude that the desired result is proved.

Sol:

We need to rewrite Euclid's fifth postulate so that it is easier to understand.
We know that Euclid's fifth postulate states that "No intersection of lines will take place when the sum of the measures of the interior angles on the same side of the falling line is exactly 180^∘.
We know that Playfair's axiom states that "For every line l and for every point P not lying on 1, there exists a unique line m passing through P and parallel to Γ.
The above mentioned Playfair's axiom is easier to understand in comparison to the Euclid's fifth postulate.
Let us consider a line l that passes through a point p and another line m. Let these lines be at a same plane.
Let us consider the perpendicular CD on l and FE on m.

From the above figure, we can conclude that CD = EF.
Therefore, we can conclude that the perpendicular distance between lines m and l will b. constant throughout, and the lines m and l will never meet each other or in other words, w can say that the lines m and I are equidistant from each other.

Sol:

Given any two distinct points A and B, there exists a third point C, which is between A and B.
There exists at least three points that are not on the same line. The undefined terms in the given postulates are point and line. The two given postulates are consistent, as they do not refer to similar situations and they refer to two different situations. We can also conclude that, it is impossible to derive at any conclusion or any statement that contradicts any well-known axiom and postulate.
The two given postulates do not follow from the postulates given by Euclid. The two given postulates can be observed following from the axiom, "Given two distinct points, there is a unique line that passes through them"