Triangles - Exercise 6.1

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Overview

Grasp the geometric core of similarity versus congruence. Understand the two essential conditions for similarity of polygons: corresponding angles must be equal, and corresponding sides must be in the same ratio (proportional).

NCERT Exercise Practice questions with detailed step-by-step solutions

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Q1: Fill Blanks Q2: Similarity Examples Q3: Pentagons Q4: Quadrilaterals Q5: Photos
Question 1 Principles of Geometric Similarity
Fill in the blanks using the correct word given in brackets:
(i) All circles are __________. (congruent, similar)
(ii) All squares are __________. (similar, congruent)
(iii) All __________ triangles are similar. (isosceles, equilateral)
(iv) Two polygons of the same number of sides are similar, if (a) their corresponding angles are __________ and (b) their corresponding sides are __________. (equal, proportional)
Detailed Answers
(i) similar — All circles have the exact same shape (roundness) regardless of radius. They are congruent only if they have the same radius.
(ii) similar — All squares have four $90^\circ$ angles and all sides equal, giving identical shapes with varying sizes.
(iii) equilateral — Every equilateral triangle has three angles equal to $60^\circ$ and equal sides, guaranteeing similarity by AAA criterion.
(iv) (a) equal, (b) proportional — By definition, two polygons are similar if and only if corresponding angles are equal and corresponding sides are proportional.
Question 2 Visual Real-Life Examples
Give two different examples of each:
(i) A pair of similar figures.
(ii) A pair of non-similar figures.
(i) Pair of Similar Figures
1. Two circles of radii $2\text{ cm}$ and $5\text{ cm}$ (same circular shape, different scale).
2. Two squares having sides $3\text{ cm}$ and $6\text{ cm}$ (both have all angles $90^\circ$ and proportional sides $\frac{3}{6} = \frac{1}{2}$).
(ii) Pair of Non-Similar Figures
1. A square of side $4\text{ cm}$ and a rhombus of side $4\text{ cm}$ (sides are equal in length, but angles are not equal).
2. A right-angled scalene triangle and an equilateral triangle (angles are completely different, so shapes differ).
Question 3 Equal Angles with Non-Proportional Sides
Two pentagons have their corresponding angles equal, but the ratios of their corresponding sides are not the same. State whether these two pentagons are similar. Give reasons for your answer.
Condition Examination
For any two polygons with $n > 3$ sides to be similar, both conditions must be satisfied simultaneously:
  1. Their corresponding angles must be equal.
  2. Their corresponding sides must be proportional (in the same ratio).
In this scenario, although the corresponding angles are equal, the corresponding sides are not in the same ratio.
No, the two pentagons are NOT similar because similarity requires both equal corresponding angles AND proportional corresponding sides.
Question 4 Proportional Sides with Unequal Angles
Two quadrilaterals have their corresponding sides in the same ratio, but their corresponding angles are not equal. Are the two quadrilaterals similar? Justify your answer.
Example Counter-Proof
Consider a square with all sides equal to $4\text{ cm}$ (all angles $90^\circ$) and a rhombus with all sides equal to $4\text{ cm}$ (having angles $60^\circ$ and $120^\circ$).
The ratios of their corresponding sides are all $\dfrac{4}{4} = 1$ (proportional).
However, their corresponding angles ($90^\circ \neq 60^\circ$) are not equal.
Therefore, they have different geometric shapes.
No, the two quadrilaterals are NOT similar because having proportional sides alone is not sufficient; corresponding angles must also be equal.
Question 5 Photographic Scaling: Similarity vs Congruence
Three photographs of the same building are taken from the same position but printed in different sizes. State whether these photographs are similar or congruent. Give reasons.
Comparison
Since all three photographs are taken of the exact same building from the exact same camera angle, the angular perspectives, outlines, and relative proportions remain identical.
However, since they are printed in different sizes:
  • They have the same shape but different sizes, which is the precise mathematical definition of similarity.
  • For figures to be congruent, they must have both the same shape and the exact same size.
The photographs are SIMILAR, but NOT CONGRUENT (identical shape, but different dimensions).
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