CBSE Class 9 Mathematics Introduction to Euclid's Geometry MCQs Set 05

Answer: (a) has no part

Explanation:
1. Euclid's first definition: a point is that which has no part.
2. Length only describes a line, length and breadth describes a surface, and lying evenly with the lines on itself describes a plane.

Teacher's Note:
1. Recall the first three definitions: point, line, ends of a line.
2. A point has no dimension.

Answer: (b) points

Explanation:
1. Euclid's third definition says that the ends of a line are points.
2. A line segment is a piece of a line with two end points.

Teacher's Note:
1. A segment AB has two end points A and B.
2. Points have no part.

Answer: (c) lines

Explanation:
1. Euclid's sixth definition: the edges of a surface are lines.
2. A surface has length and breadth only, and its boundaries are lines.

Teacher's Note:
1. Think of the edges of a sheet of paper.
2. Edges have length but no breadth.

Answer: (b) all right angles are equal to one another

Explanation:
1. The fourth postulate: all right angles are equal to one another.
2. Every right angle measures \( 90^\circ \).

Teacher's Note:
1. It gives a standard unit for comparing angles.
2. Right angles are all 90 degrees.

Answer: (c) infinite

Explanation:
1. Through a given point infinitely many lines can be drawn, in all directions.
2. It is only through two distinct points that exactly one line passes.

Teacher's Note:
1. Think of spokes of a wheel at its hub.
2. Compare with the case of two given points.

Answer: (c) collinear points

Explanation:
1. Points are collinear if there is a line that contains all of them.
2. Concurrent applies to lines: three or more lines through one point.

Teacher's Note:
1. Co- means together, linear refers to a line.
2. Points P, Q, R, S, T on a line \( \ell \) are collinear.

Answer: (d) concurrent lines

Explanation:
1. Lines are concurrent if there is a point lying on all of them.
2. Collinear is the word used for points, not lines.

Teacher's Note:
1. Think of the lines meeting at a common point P.
2. Parallel lines never meet.

Answer: (a) \( R \) lies between \( P \) and \( Q \) and is neither \( P \) nor \( Q \)

Explanation:
1. An interior point lies strictly between the end points.
2. End points themselves are not interior points.

Teacher's Note:
1. The mid-point is an example of an interior point.
2. Points beyond the ends are outside the segment.

Answer: (b) they have the same length

Explanation:
1. Segments \( AB \) and \( CD \) are congruent when a trace copy of one covers the other exactly, i.e. when their lengths are equal.
2. Position or direction does not matter.

Teacher's Note:
1. Congruent means identical in size and shape.
2. We write AB is congruent to CD.

Answer: (c) the length of the line segment \( PQ \)

Explanation:
1. Distance between two points is the length of the segment joining them.
2. A whole line has no finite length.

Teacher's Note:
1. Measure along the segment.
2. A line extends without end.

Answer: (a) length and breadth only

Explanation:
1. Euclid's fifth definition: a surface is that which has length and breadth only.
2. Length only describes a line.

Teacher's Note:
1. A surface is two-dimensional.
2. A solid adds height.

Answer: (b) that which has length only

Explanation:
1. Euclid's definition of a straight line: that which has length only (a line is breadthless length).
2. A point is that which has no part, and the edges of a surface are lines.

Teacher's Note:
1. A straight line has one dimension.
2. The two phrases 'breadthless length' and 'length only' express the same idea.