Sure Shot Questions for Board Exams: Introduction to Euclid's Geometry

Ans:
Here, BX = 1/2 AB and BY = 1/2 BC ...(i) [given]
Also, AB = BC [given]
⇒ 1/2AB = 1/2BC ...(ii)
[∵ Euclid's seventh axiom says, things which are halves of the same thing are equal to one another]
From (i) and (ii), we have
BX = BY

Ans:
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

Ans:
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.

Ans:
(i) Four collinear points are D, E, F, G and H, I, J, K
(ii) Five rays are DG, EG, FG, HK, IK.
(iii) Five line segments are DH, EI, FJ; DG, HK.
(iv) Two-pairs of non-intersecting line segments are (DH, EI) and (DG, HK).

Ans:
Given, AC = BC
Now, add AC on both sides.
L.H.S + AC = R.H.S + AC
AC + AC = BC + AC
2AC = BC + AC
Since, we know,
BC +AC = AB (as it coincides with line segment AB, from figure)
∴ 2 AC = AB (If equals are added to equals, the wholes are equal.)
⇒ AC = 1/2 AB.

Ans:
It is given, AC = BD
From the given figure, we get,
AC = AB + BC
BD = BC + CD
⇒ AB + BC = BC + CD [Given: AC=BD]
We know that, according to Euclid's axiom, when equals are subtracted from equals, remainders are also equal.
Subtracting BC from the L.H.S and R.H.S of the equation AB + BC = BC + CD, we get,
AB + BC - BC = BC + CD - BC
AB = CD
Hence proved.

Ans:
According to the question,
We have,
x+y=10 ...(i)
And, x=z ...(ii)
Applying Euclid's axiom,
"if equals are added to equals, the wholes are equal"
We get,
From Eq. (i) and (ii)
x+y=z+y ....(iii)
From Eqs. (i) and (iii)
z+y = 10

Ans: 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.

Ans: Euclid's postulates were:

• A straight line may be drawn from one point to any other point.
• A terminated line can be produced indefinitely.
• A circle can be drawn with any centre and any radius.
• All right angles are equal to one another.
• 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.

Ans: 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