Euclid's Geometry — Chapter Test | Class 9 IMO

Q1

Given three line segments AB, CD, and EF. If AB coincides with CD, and CD coincides with EF, then AB coincides with EF. This means AB = EF. Which axiom handles the transitivity of coinciding items?

Axiom 4 states that things which coincide are equal. Since AB coincides with CD, AB = CD. Since CD coincides with EF, CD = EF. Then Axiom 1 states that things equal to the same thing are equal to one another, so AB = EF.

Q2

If a point X lies inside an angle ∠AOB, then the measure of ∠AOB is always ________ the measure of ∠AOX.

Since X lies inside the angle, ∠AOX is a part of the whole angle ∠AOB. According to Euclid's Fifth Axiom, the whole is greater than the part, so ∠AOB > ∠AOX.

Q3

An equilateral triangle can be constructed on any given line segment. This proposition was proven by Euclid using which of his tools?

In Proposition 1 of Book 1, Euclid constructed an equilateral triangle on a segment AB by drawing two circles centered at A and B with radius AB (Postulate 3), finding their intersection point via lines (Postulate 1), and equating the sides using Axiom 1.

Q4

If two straight lines l and m are intersected by a transversal n such that the sum of interior angles on one side is exactly 179°, then what happens if the lines are extended on that side?

By Euclid's Fifth Postulate, since the sum of the interior angles (179°) is less than two right angles (180°), the lines will intersect/meet if produced indefinitely on that side.

Q5

A sheet of paper has length and breadth. It represents a surface. According to Euclid, the boundaries of this sheet of paper are:

The boundaries of a surface are lines. The edges of a sheet of paper form lines.

Q6

Two distinct lines l and m intersect at a point O. Can they intersect at another point P as well?

A fundamental theorem derived from Euclid's axioms states that 'Two distinct lines cannot have more than one point in common.' Thus they can intersect at only one point.

Q7

The flat surface of a highly polished standard wooden study table can be modeled geometrically as a:

A flat surface that extends thin and flat is modeled as a plane or plane surface in geometry.

Q8

An ancient Indian text containing rules for the construction of sacrificial altars and geometric structures is called:

The Sulba Sutras are ancient Indian mathematical texts that contain detailed geometric instructions for designing and constructing various sacrificial altars.

Q9

If a point C lies between two points A and B such that AC = BC, then which of the following is true regarding AC and AB?

We are given AC = BC. According to Euclid's axiom, if equals are added to equals, the wholes are equal. Add AC to both sides: AC + AC = BC + AC. This gives 2 AC = AB (since C lies between A and B, AC + BC coincides with AB). Therefore, AC = ½ AB.

Q10

Two students are given identical lengths of strings, S1 = S2. Student 1 cuts off a piece of 5 cm from S1, and Student 2 cuts off a piece of 5 cm from S2. The remaining pieces are L1 and L2. Which of the following is a valid mathematical deduction?

Initial lengths are equal (S1 = S2). Equal lengths (5 cm) are subtracted from both. According to Euclid's Third Axiom, the remainders are equal, so L1 = L2.

Q11

In spherical geometry (the geometry of the surface of a sphere), a 'straight line' is represented by a great circle. In this system, the sum of angles of a triangle is:

Spherical geometry is a type of non-Euclidean geometry. On a sphere, the sides of a triangle bulge outwards, making the sum of the interior angles strictly greater than 180°.

Q12

Which Greek mathematician is famous for stating that the world is made up of points, lines, and planes, and taught Euclid's predecessors that geometry requires deductive proofs?

Thales of Miletus was the first Greek mathematician who initiated deductive geometry and provided the first formal proofs.

Q13

If AB = CD, then which of the following expressions is true when we add BC to both sides?

Given AB = CD. Adding BC to both sides (Euclid's Second Axiom): AB + BC = CD + BC. Since B lies between A and C, AB + BC = AC. Since C lies between B and D, CD + BC = BD. Therefore, AC = BD.

Q14

Pythagoras was a famous student of which mathematician?

Pythagoras was a pupil of Thales, and he along with his followers discovered many geometric properties.

Q15

If two lines intersect in such a way that the measure of one of the angles is 90°, then all four angles formed at the intersection are right angles and are equal to each other. Which postulate underlies the equality of all these right angles?

Euclid's Fourth Postulate states that all right angles are equal to one another, providing the foundation for equating right angles regardless of their position.