Ch 5: Euclid's Geometry - Exercise 5.1 Practice

Q1: True or False

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State whether the following statements are true or false. Give reasons for your answers.
(i) A surface has length and breadth only.
(ii) The edges of a surface are lines.
(iii) If two circles are equal, then their radii are unequal.

(i) True: Euclid defined a surface as that which has length and breadth only.

(ii) True: The boundaries of a surface are curves or straight lines.

(iii) False: If two circles are equal (congruent), their areas and circumferences are equal, which implies their radii must be equal.

(i) True, (ii) True, (iii) False

Q2: Definitions

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Define the following terms:
(i) Intersecting Lines
(ii) Concurrent Lines
(iii) Ray

(i) Intersecting Lines: Two lines are said to be intersecting if they have a common point. The common point is called the point of intersection.

(ii) Concurrent Lines: Three or more lines in a plane are said to be concurrent if all of them pass through the same point.

(iii) Ray: A part of a line with one endpoint (start point) and extending infinitely in one direction is called a ray.

Q3: Geometric Proof

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If a point $C$ lies between two points $A$ and $B$ such that $AC = BC$, then prove that $AC = \frac{1}{2}AB$. Explain by drawing the figure.

Given: $AC = BC$.

Add $AC$ to both sides (Euclid's Axiom: If equals are added to equals, the wholes are equal).
$AC + AC = BC + AC$

$2AC = AB$ (Since $BC + AC$ coincides with $AB$).

Therefore, $AC = \frac{1}{2}AB$.

Proved.

Q4: Axiom Application

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In a figure where points $A, B, C, D$ lie on a line in that order, if $AC = BD$, prove that $AB = CD$.

Given: $AC = BD$.

From the figure, $AC = AB + BC$ and $BD = BC + CD$.

Substituting these values: $AB + BC = BC + CD$.

Subtracting $BC$ from both sides (Euclid's Axiom: If equals are subtracted from equals, the remainders are equal):
$AB = CD$.

Proved.