CBSE Class 9 Mathematics Introduction to Euclid's Geometry MCQs Set 07

Answer: (b) False

Explanation:
1. By Euclid's first postulate, a straight line may be drawn from any one point to any other point.
2. Moreover, only one line passes through two distinct points.
3. So the statement that many lines pass through them is false.

Teacher's Note:
1. Two distinct points determine a unique line.
2. Infinitely many lines pass through one point, but only one through two points.

Answer: (a) True

Explanation:
1. A system of axioms is consistent if no statement can be proved both true and false from it.
2. Any axiomatic system must have this property, so the statement is true.

Teacher's Note:
1. Consistency means that the axioms never contradict each other.
2. This is a basic requirement of any deductive system.

Answer: (d) (i), (iii) and (iv)

Explanation:
1. Lines are parallel when they have the same direction (same slope).
2. Lines (i), (iii) and (iv) all slope upwards at the same angle, so they are parallel.
3. Line (ii) slopes in the opposite direction, so it is not parallel to the others.

Teacher's Note:
1. Parallel lines never meet and keep the same direction.
2. Compare the slopes in the figure.

Answer: (a) mid-point of PR

Explanation:
1. A point lying between two end points of a segment and equidistant from both is the mid-point.
2. So Q is the mid-point of PR.

Teacher's Note:
1. The mid-point divides a segment into two equal parts.
2. A segment has one and only one mid-point.

Answer: (d) PS = QR

Explanation:
1. From the figure, PQ = PS + SQ and SR = SQ + QR.
2. Given PQ = SR, so PS + SQ = SQ + QR.
3. Subtracting the equal part SQ from both sides (equals subtracted from equals are equal) gives PS = QR.

Teacher's Note:
1. Use the common notion that equals subtracted from equals give equals.
2. Write each long segment as the sum of its small parts.

Answer: (a) True

Explanation:
1. If a segment had two mid-points, the two points would be at the same distance from an end point, which forces them to coincide.
2. So every line segment has exactly one mid-point.

Teacher's Note:
1. This is a theorem that follows from the axioms.
2. Halves of the same thing are equal.

Answer: (a) True

Explanation:
1. An angle is formed by two rays that have a common initial point, which is the vertex.
2. So the statement is true.

Teacher's Note:
1. The common end point is called the vertex of the angle.
2. The rays are the arms of the angle.

Answer: (a) True

Explanation:
1. Points that lie on one straight line are collinear by definition, so the statement is true.

Teacher's Note:
1. Collinear means on the same line.
2. Any two points are always collinear.

Answer: (a) True

Explanation:
1. This is Euclid's third postulate, so the statement is true.

Teacher's Note:
1. Compass constructions depend on this postulate.
2. A circle is fixed by its centre and radius.

Answer: (b) False

Explanation:
1. Euclid's first postulate says that a straight line may be drawn from any one point to any other point.
2. The given statement says the opposite, so it is false.

Teacher's Note:
1. Read the word "not" carefully.
2. The postulate guarantees a line between any two points.

Answer: (a) True

Explanation:
1. Two circles are equal (congruent) when they coincide on superposition.
2. Then their centres coincide and the radii must match, so the radii are equal.

Teacher's Note:
1. Things which coincide with one another are equal to one another.
2. Equal circles have equal radii and equal diameters.

Answer: (b) equal to one another

Explanation:
1. This is Euclid's fourth common notion (axiom): things which coincide with one another are equal to one another.

Teacher's Note:
1. This is the idea behind the superposition method for proving congruence.
2. Learn the seven common notions.

Answer: (a) assumptions

Explanation:
1. Axioms and postulates are assumptions taken as obvious universal truths and are not proved.
2. Theorems are proved, so they cannot fill the blank.

Teacher's Note:
1. Axioms are used throughout mathematics, postulates are specific to geometry.
2. They are accepted without proof.

Answer: (a) remainders

Explanation:
1. Euclid's third common notion says that if equals are subtracted from equals, the remainders are equal.

Teacher's Note:
1. This is used to compare parts of segments.
2. If a = b then a - c = b - c.

Answer: (a) right

Explanation:
1. Euclid's fourth postulate states that all right angles are equal to one another.

Teacher's Note:
1. Every right angle measures 90 degrees.
2. This postulate gives a fixed standard of angle.

Answer: (a) point

Explanation:
1. Euclid defined a point as that which has no part, so it has no length, breadth or thickness.

Teacher's Note:
1. A point only shows position.
2. A line is breadthless length.

Answer: (a) points

Explanation:
1. Euclid defined the ends of a line as points.
2. The edges of a surface are lines.

Teacher's Note:
1. Solids have surfaces as boundaries, surfaces have lines and lines have points.
2. This is the chain of dimensions.

Answer: (a) greater than

Explanation:
1. Euclid's fifth common notion says that the whole is greater than the part.

Teacher's Note:
1. This holds for finite quantities.
2. A segment is longer than any of its parts.

Answer: (a) halves

Explanation:
1. Euclid's seventh common notion says that things which are halves of the same things (or double of the same thing) are equal to one another.
2. So halves fill the blank.

Teacher's Note:
1. Two halves of the same quantity are equal.
2. The same holds for doubles.

Answer: (a) postulates

Explanation:
1. Euclid called the assumptions specific to geometry postulates.
2. The general assumptions used throughout mathematics are the common notions or axioms.

Teacher's Note:
1. Postulates are geometric and axioms are general.
2. Both are accepted without proof.

Answer: (a) parallel

Explanation:
1. This follows from Playfair's axiom: through a point not on a given line, only one line can be drawn parallel to that line.
2. If two intersecting lines were both parallel to one line, two parallels would pass through their common point, which is impossible.

Teacher's Note:
1. Playfair's axiom is equivalent to Euclid's fifth postulate.
2. Intersecting lines have a common point.