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**Worksheet (Part - 1)**

**1.** If A, B and C are three points on a line, and ‘B’ lies between ‘A’ and ‘C’ (as shown in the figure), then prove that: AB + BC = AC

**2.** Prove that an equilateral triangle can be constructed on any given line segment.

**3.** Prove that two distinct lines cannot have more than one point in common.

**4.** Is the following statement a direct consequence of Euclid’s fifth postulate? “There exists a pair of straight lines that are everywhere equidistant from one another.”**Hint:** Use playfair’s axiom, which is equivalent to Euclid’s fifth postulate.

**5. **Is the following statement true? “Attempts to prove Euclid’s fifth postulate using the other postulate and axioms led to the discovery of several other geometries.”**ANSWERS****4.** Yes

**5.** Yes**Worksheet (Part - 2)**

**1.** Fill in the blanks to complete the following axioms :

(i) Things, which are equal to the same things, are ...............................

(ii) If equals are added to equals, the ...............................

(iii) If equals are subtracted from equals, ...............................

(iv) Things which coicide with one another are ...............................

(v) The whole is greater than the ...............................

(vi) Things which are double of the same things are ...............................

(vii) Things which are halves of the same things, are ...............................

(viii) If equals are multiplied by equals, then their ...............................

(ix) If equals are divided by equals, then their ...............................

(x) Of the two quantities of the same kind, the first is greater than, equal to or less than the second. This axiom is called ...............................**2.** In the figure, line PQ falls on AB and CD such that (∠1 + ∠2) < 180°. So, lines AB and CD, if produced will intersect on the left of PQ. This is an example of which Postulate of Euclid?

**3. **In the given figure, if AC = BD, then prove that AB = CD**Hint:** AC = BC [given] ...(1)

∴ AC = AB + CD [∵ B lies between A and C] ...(2)

BD = BC + CD [∵ C lies between B and D] ...(3)

From (1), (2) and (3) we have: AB + BC = BC + CD or AB = CD**4.** Write ‘true’ or ‘false’ for the following statement:

(i) Three lines are concurrent if they have a common point.

(ii) A line separates a plane into three parts, namely the two half planes and the line itself. (iii) Two distinct lines in a plane cannot have more than one point in common.

(iv) A ray has two end points.

(v) A line has indefinite length.**5. **How many lines can pass through:

(i) one point

(ii) two distinct points?**6.** AB and CD are two distinct lines. In how many points can they at the most intersect?

**7.** Prove that any line segment has one and one mid-point.

**8.** If P, Q and R are three points on a line and Q lies between P and R, then show that PQ + QR = PR

**Hint: **Use the axiom – 4 that “things which coincide with one another are equal to one another.”**9.** In the given figure, AB = BC, P is mid point of AB and Q is mid point of BC. Show that AP = QC**Hint: **Things which are halves of the same thing (or equal things) are equal to one another.**ANSWERS****1.** (i) equal to one another

(ii) wholes are equal

(iii) the remainders are equal

(iv) equal to one another

(v) part

(vi) equal to one another

(vii) equal to one another

(viii) products are equal

(ix) quotients are equal

(x) ‘Trichotomy law’

**2.** Fifth postulate

**4.** (i), (ii), (iii) and (v) are true.

**5.** (i) infinite

(ii) one only

**6.** one point only

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