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Q1. Which one of the following statement is true?

• 1) Two distinct lines cannot have more than one point in common
• 2) There are an infinite number of lines which pass through two distinct points.
• 3) If two circles are equal, then their radii are not equal.
• 4) Only one line can pass through a single point.

Solution

Statement (A) is incorrect as through a single point infinite number of lines can pass. Statement (B) is incorrect as through 2 distinct points, one and only one line can pass which contains both the points. Statement (D) is incorrect as, if the 2 circles are equal then their radii are also equal. Statement (C) is correct i.e. 2 distinct lines cannot have more than one point in common.

Q2. How many points can be common in two distinct straight lines?

• 1) one
• 2) None
• 3) two
• 4) three

Solution

Two distinct lines can intersect only at one point.  Hence there can be only one point common to the two distinct lines.  

Q3. The things which are double of same things are:

• 1) Equal
• 2) Unequal
• 3) double of the same thing
• 4) halves of same thing

Solution

Q4. A proof is required for:

• 1) Axiom
• 2) Definition
• 3) Postulate
• 4) Theorem

Solution

Theorem

Q5. 'Two intersecting lines cannot be parallel to the same line' is stated in the form of:

• 1) An axiom
• 2) A definition
• 3) A proof
• 4) A postulate

Solution

A postulate

Q6. The number of line segments determined by three collinear points is:

• 1) Three
• 2) Four
• 3) Only one
• 4) Two

Solution

Three non-collinear points determines three line segments.

Q7. A surface is that which has

• 1) Breadth only
• 2) Length only
• 3) Length and height
• 4) Length and breadth

Solution

A surface is that which has both length and breadth.

Q8. If the point P lies in between M and N and C is midpoint of MP then:

• 1) MC + PN = MN
• 2) CP + CN = MN
• 3) MC + CN + MN
• 4) MP + CP = MN

Solution

MC + CN = MN

Q9. 'Lines are parallel if they do not intersect' - is stated in the form of:

• 1) A proof
• 2) A definition
• 3) A postulate
• 4) An axiom

Solution

A definition

Q10. How does Euclid's fifth postulate imply the existence of parallel lines? Give a mathematical proof.

Solution

Euclid's 5^th postulate states that: 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. This implies that if n intersects lines l and m and if . In that case, producing line l and m further will meet in the side of1 and 2 which is less than 180^o If In that case, the lines l and m neither meet at the side of 1 and 2 nor at the side of 3 and 4. Implying that the lines l and m will never intersect each other. Therefore, the lines are parallel.

Q11. Define: (A) line segment (B) radius of a circle

Solution

(A) A line segment is a part of line between two points. (B) Radius of a circle is defined as the distance between center and a point on its circumference.

Q12. Define : (A) a square (B) Perpendicular lines

Solution

Square - A square is a rectangle with a pair of consecutive sides equal. Perpendicular lines - Two lines are perpendicular, if the angle between them is 90^o.

Q13. In figure, AC= XD, C is midpoint of AB and D is midpoint of XY. Using a Euclid's axiom, show that AB= XY.

Solution

AB = 2AC (C is midpoint of AB) XY = 2XD (D is midpoint of XY) Also AC = XD (Given) Therefore, 2AC = 2 XD Hence, AB = XY because things which are double of same things are equal to one another.

Q14. Which of the following is an example of a geometrical line?

• 1) Black Board
• 2) Tip of the sharp pencil
• 3) Meeting place of two walls
• 4) Sheet of paper

Solution

Meeting place of two walls is an example of a geometrical line.

Q15. Prove that every line segment has one and only one midpoint.

Solution

Suppose C and C' are two mid points of segment AB Then AC = AB and AC' = AB AC = AC' [Things which are equal to the same thing are equal to one another.] This is possible only when C and C' coincide. Hence every line segment has one and only one midpoint.

Q16.

Solution

(i) Only one(ii) Only one(iii) NoneBecause, the axiom to Euclid's Postulate 2 states that: Given two distinct points, there is a unique line that passes through them.

Q17. If a point C lies between two points A and B such that AC = BC, then prove that AC = AB. Explain by drawing the figure.

Solution

Given :AC = BC Add AC to both sides AC + AC = BC + AC (If equals are added to equals, the wholes are equal) 2AC = AB Hence, AC = AB

Q18. In figure, if PS = RQ then prove that PR = SQ.

Solution

In fig., we have PS = RQ PR + RS = RS + SQ By subtracting, RS, we get PR = SQ Hence proved.

Q19. If x > y then, there exists a z such that x = y + z. What is the condition on z? State the postulate to which this statement corresponds.

Solution

Q20.

Solution

Let l and m be two lines intersecting at point A Let us suppose they intersect at one more point Bi.e l and m pass through two distinct points A and B.But, the Euclid's Postulate 2 states that: Given two distinct points, there is a unique line that passes through them. So, our supposition is incorrectTherefore, two distinct lines can not have more than one point in common.

Q21. By using the Euclid's postulates and Axioms, show that two lines perpendicular to the same lines are parallel to each other.

Solution

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