Introduction to Euclid's Geometry Important Questions

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Questions:

Generic 3

4 x 3 = 12

In the given figure, we have \(\angle1=\angle3\) and \(\angle2=\angle4\). Show that, \(\angle A=\angle C.\)

In a triangle ABC, X and Y are the points On AB and BC such that BX = BY and AB = BC. Show that AX = CY. State the Euclid's Axiom used.

Does Euclid 's fifth postulate simply the existence of parallel lines? Explain.

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.

Generic 4

5 x 4 = 20

Read the following statement :
"A square is a polygon made up of four line segments, out of which, length of three line segement, out of which length of three line segments are equal to the length of fourth one and all its angles are right angles."
(i) Define the terms used in this definition which you feel necessary. Are there any undefined terms in this? Can you justify that all angles and sides of a square are equal?
(iii) What is its value?

In the fig., if \(OX=\frac{1}{2}XY,\ PX=\frac{1}{2}XZ\) and OX = PX, Show that XY = XZ. State which axiom you use here. Also give two more axioms other than the oxiom used in the above situation.

(i) If a point C lies between two points A and B such that AC = BC, then prove that AC= AB.
(ii) Is CB = \(1\over2\) AB?
(iii) Apala says that the ratio of AC and BC is 1 : 1. Is she correct? If so, which value of Apala is depicted by her statement?
(iv) Which mathematical concept has been covered in this problem?
(v) Write the formulae used in the solution

In the given figure, it is given that \(\angle \)1 = \(\angle \)4 and \(\angle \)3 = \(\angle \)2. By which Euclid's axiom, it can be shown that if \(\angle \)2 = \(\angle \)4, then \(\angle \)1 = \(\angle \)3.

In the given figure, if AB = CD, then prove that AC = BD. Also, write the Euclid's Axiom used for proving it.

Multiple Choice Question

7 x 1 = 7

10.

There exists a pair of straight lines that are everywhere equidistant from one another' is a direct consequence of Euclid's

(a)

first postulate

(b)

second postulate

(c)

third postulate

(d)

fifth postulate

11.

Two interesting lines cannot be parallel to the same line, is started in the form of:

(a)

an axiom

(b)

a definition

(c)

a postulate

(d)

a proof

12.

Given four points such that no three of them are collinear, then the number of lines that can be drawn through them is:

(a)

2 lines

(b)

4 lines

(c)

6 lines

(d)

8 lines

13.

Which one of the following statements is true?

(a)

Only one line can pass through a single point.

(b)

Three are an infinite number lines which pass through two distinct points.

(c)

two distinct lines cannot have more than one point in common.

(d)

If two circles are equal, then their radii are not equal.

14.

Euclid stated that things which are equal to the same thing are equal to one another in the form of:

(a)

an axiom

(b)

a definition

(c)

a postulate

(d)

a proof

15.

The number of lines that can pass through a given point is:

(a)

two

(b)

none

(c)

only one

(d)

infinite many

16.

Pythagoras was a student of

(a)

Thales

(b)

Euclid

(c)

both (a) and (b)

(d)

Archimedes

1 Marks

3 x 1 = 3

17.

How can we identify parallel lines?

18.

What is a straight line?

19.

What does a theorem require?

Generic 2

5 x 2 = 10

20.

Prove that every line segment has one and only one mid-point.

21.

State playfair's axiom. Is it equivalent to one of the Euclid's postulate.

22.

How many planes can be made to pass through
(i) Three collinear points.
(ii) Three non-collinear points.

23.

State any two Euclid's axioms.

24.

State any two Eulis's axioms.

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Answers Key:

Generic 3

4 x 3 = 12

Since \(\angle1=\angle3\ and \ \angle2=\angle4\), therefore adding before equations.
\(\angle1+\angle2=\angle3+\angle4\)
\(\Rightarrow \angle BAD= \angle BCD\)
\(\Rightarrow \angle A= \angle C.\)

AB = BC (given)
BX = BY (given)
If equals are subtracted from equals, then remains are also equal.
AB - BX = BC - BY
\(\Rightarrow\) AX = CY

If a straight line I falls on two straight lines m and n such that sum of the interior angles on one side of I is two right angles, then by Euclid's fifth postulate the lines m and n will not meet on this side of I. Next, we know that the sum of the interior angles on the other side of line I will also be two right angles. Therefore, they will not meet on the other side also. So, the lines m and n never meet and are, therefore arallel.

Yes! These postulates contain two undefined terms: Point and Line. Yes! These postulates are consistent because they deal with two different situations
(i) says that given two points A and B, there is a point C lying on the line in between them,
(ii) says that given A and B, we can take C not lying on the line through A and B. These 'postulates' do not follow from Euclid's postulates, however, they follow from Axiom 'Given two distinct lines, there is a unique line that passes through them.

Generic 4

5 x 4 = 20

(i) The terms need to be defined are:
Polygon: A simple closed figure made up of three or more line segments.
Line segment: part of a line with two end points.
line: undefined term.
Point: undefied term.
Angle: A figure formed by two rays with a common initial point.
Ray: Part of a line with one end point.
Right angle: Angle whose measure is 90^o
Undefined terms used are: Line, point
Euclid's fourth postulate says that "all right angles are equal to one another." in a square, all angles are right angles, therefore all angles are equal (From Euclid's fourth postulate)
Three line segments are equal to fourth line segment.
Therefore, all the four sides of a square are equal( by Euclid's first axiom " things which are equal to one another")
(ii) Introduction to Euclid's geometry
(iii) Equality leads to democracy.

Here \(OX=\frac{1}{2}XY\)
\(PX=\frac{1}{2}XZ\)
Also OX = PX
\(\Rightarrow \frac{1}{2}XY=\frac{1}{2}XZ\)
Things equal to half of equals, are equal to one another.
Two other axioms:
Things coincide with one another are equal to one another.
e.g., If \(\overline{AB}\) coincide with \(\overline{XY}\), such that A falls on X and B falls on Y, then \(\overline {AB}=\overline{XY}\)
The whole is greater than the part
e.g., if \(m\angle1=m\angle2+m\angle3,\ then\ m\angle1>m\angle2\ and\ m\angle1>m\angle3\)

AC = BC I Given

⇒ AC + AC = BC + AC | If equals are added to equals, the wholes are equal.
⇒ 2AC = AB
AC = \(1\over2\) AB
(ii) AC = OC I Given
AC = \(1\over2\)AB I Proved in (i) above
\(\therefore\) CB = \(1\over2\) AB
(iii) AC = BC I Given
\(\therefore\) AC: BC = 1 : 1
\(\therefore\) Apala is correct. So, the value 'Sharpness' is depicted by her statement.
(iv) The mathematical concept 'Introduction to Euclid's Geometry' has been covered in this problem.
(v) The formulae used in the solution are as follows:
1. If equals are added to equals, the wholes are equal.
2. Concept of ratio.

\(\angle \)2 = \(\angle \)4 ,,,,,,,(1) | Given
\(\angle \)1= \(\angle \)4 ........(2) | Given
\(\angle \)3 =\(\angle \)2 .......(3) | Given
From (1), \(\angle \)4 = \(\angle \)2 ......(4)
From (3) and (4),
\(\angle \)3 = \(\angle \)4 .....(5)
I Things which are equal to the same thing are equal to one another
From (2) and (4),
\(\angle \)1 = \(\angle \)3
I Things which are equal to the same thing are equal to one another

We have, AX=CY
AB=CD
\(\Rightarrow \) AB + BC = CD + BC
| If equals are added to equals, the wholes are equal (Euclid's Axiom (ii))
AC= BD
I AB + BC coincides with AC;
CD + BC coincides with BD
[Things which coincide with one another are equal to one another (Euclid's Axiom (iv))]

Multiple Choice Question

7 x 1 = 7

10.

(d)
fifth postulate

11.

12.

13.

14.

(a)
an axiom

15.

(d)
infinite many

16.

(a)
Thales

1 Marks

3 x 1 = 3

17.

()
Lines are parallel if they do not intersect on being extended.
For example:

Lines A and B are parallel lines.

18.

()
Two planes intersect each other to form a straight line.

19.

()
Theorem requires a proof.

Generic 2

5 x 2 = 10

20.

Let line segment \(\overline{AB}\) has 2 mid-points, say X and Y
then, \(\frac{AB}{2}=AX\ and\ \frac{AB}{2}=AY\)
\(\therefore\) AX = AY
X and Y coincides

21.

Playfair's Axiom(Statement) : For every line l and for every point P not lying on l, there exists a unique line m passing through P and parallel to l. it is equivalent to Euclid's fifth postulate.

22.

(i) Infinite, if they are collinear.
(ii) Only one, if they are non-collinear points.

23.

Euclid's axioms
(i) Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equal.

24.

Euclid's axioms
(i) Things which are equal to the same thing are equal to one another.
(ii) If equals are added to equals, the wholes are equal.