TRIANGLES-Exercise 6.1

The chapter Triangles stands at the heart of Class X geometry, introducing learners to profound mathematical ideas such as similarity, proportionality, and the elegant structure underlying geometric figures. These NCERT solutions are carefully crafted to guide students through each exercise with clarity, precision, and conceptual insight. Rather than focusing only on final answers, the solutions emphasize the reasoning process—helping learners connect theorems, identify relationships, and recognize patterns that define the geometry of triangles. Every theorem, whether it is the Basic Proportionality Theorem (Thales’ Theorem), the criteria for similarity, or the area relations of similar triangles, has been interpreted in a step-by-step, student-friendly manner. The aim is to build confidence, encourage analytical thinking, and strengthen foundational understanding so learners can approach board examinations with clarity and accuracy. These solutions serve as a reliable companion for revision, practice, and problem-solving mastery, ensuring that students gain a deep and lasting comprehension of this essential chapter of NCERT Mathematics.

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December 5, 2025 | By Academia Aeternum

TRIANGLES-Exercise 6.1

Maths - Exercise

Q1. 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)

Solution:

All circles are similar (congruent, similar).

Explanation: Circles are all similar because they have the same shape and their radii differ by a scale factor, but they are not necessarily congruent unless the radii are equal.
All squares are similar (similar, congruent).

Explanation: All squares are similar as all have equal angles and their sides are proportional, but they are congruent only when their sides are exactly equal.
All triangles are similar if their corresponding angles are equal (isosceles, equilateral).

Explanation: Triangles are similar when their corresponding angles are equal, regardless of side lengths. The types of triangles like isosceles or equilateral relate to side lengths and angles but similarity depends on angle equality.
Two polygons of the same number of sides are similar, if (a) their corresponding angles are equal and (b) their corresponding sides are proportional (equal, proportional).

Explanation: For two polygons with the same number of sides to be similar, each pair of corresponding angles must be equal, and the lengths of corresponding sides must be proportional to each other.

Q2. Give two different examples of pair of
(i) similar figures.
(ii) non-similar figures.

Solution:

Q2. Give two different examples of pair of
1. similar figures:
  - A rectangle measuring 6 cm by 4 cm and another rectangle measuring 9 cm by 6 cm (both have opposite sides equal and all angles \(90^\circ\), with sides proportional by scale factor \(\frac{3}{2}\)).
  - An equilateral triangle with side 5 cm and another equilateral triangle with side 8 cm (all angles \(60^\circ\) in both, sides proportional by scale factor \(\frac{8}{5}\)).
2. non-similar figures:
  - A square with side 5 cm and a rectangle with sides 6 cm by 4 cm (square has all sides equal and angles \(90^\circ\), but rectangle's unequal adjacent sides make proportions differ).
  - A right-angled triangle with sides 3 cm, 4 cm, 5 cm and an equilateral triangle with side 4 cm (right triangle has \(90^\circ\) angle, while equilateral has all \(60^\circ\) angles).

Q3. State whether the following quadrilaterals are similar or not:

Solution:

The given quadrilaterals are not similar.

The first quadrilateral PQRS is a rhombus (all sides 1.5 cm but angles are not \(90^\circ\)).
The second quadrilateral ABCD is a square (all sides 3 cm and all angles \(90^\circ\)).

Although their corresponding sides are in the same ratio \(\frac{1.5}{3}=\frac{1}{2}\), their corresponding angles are not equal, so the condition for similarity of polygons (equal angles and proportional sides) is not satisfied.

Frequently Asked Questions

A triangle is a closed figure formed by three line segments and has three vertices, three sides, and three angles.

When two triangles have the same shape and size, their corresponding sides and angles are equal; they are said to be congruent.

The main congruence rules are SSS, SAS, ASA, AAS, and RHS for right triangles.

Two triangles are similar if their corresponding angles are equal and corresponding sides are in proportion.

AAA / AA, SAS similarity, and SSS similarity.

If two angles of one triangle are equal to two angles of another, the triangles are similar.

If a line is drawn parallel to one side of a triangle to intersect the other two sides, it divides the sides proportionally.

Thales’ Theorem is another name for the Basic Proportionality Theorem (BPT).

If a line divides any two sides of a triangle in the same ratio, the line must be parallel to the third side.

In a right-angled triangle: \(a^2 + b^2 = c^2\), where \(c\) is the hypotenuse.

If for a triangle \(a^2 + b^2 = c^2\), the triangle is right-angled.

The ratio of areas of two similar triangles is equal to the ratio of the squares of their corresponding sides.

The sides and angles that occupy the same relative position in congruent or similar triangles.

By showing the ratio of all three pairs of corresponding sides is equal.

If the hypotenuse and one side of a right-angled triangle are equal to the hypotenuse and one side of another right-angled triangle, the triangles are congruent.

Used in map-making, architecture, engineering, shadow measurement, surveying, and scaling models.

The ratio of corresponding sides of similar triangles.

Corresponding sides are in proportion.

\(\frac{Area_1}{Area_2} = \left(\frac{side_1}{side_2}\right)^2\).

Perimeters are in the same ratio as corresponding sides.

Yes. Similarity requires same shape, not same size.

It means the ratios of corresponding sides are equal.

The line joining midpoints of two sides of a triangle is parallel to the third side and half of it.

It is a specific case of BPT where each side is divided in the ratio 1:1.

Helps in dividing lines proportionally and constructing parallel segments.

Look at the relative position of vertices in both triangles.

Being at equal distance from two or more points/lines.

A triangle with one angle equal to \(90^\circ\).

The longest side is hypotenuse; the other two are legs or perpendicular and base.

Trigonometric ratios are defined based on similar right triangles, making ratios consistent.

Helps to calculate the distance between two points.

Yes, but triangles are simplest because if two triangles have two equal angles, the third automatically matches.

A geometric tool used to divide lengths in fixed ratios, often based on triangle similarity.

Used in navigation, construction, height-distance problems, physics, and engineering.

By drawing a line parallel to one side, forming two similar triangles, then equating ratios of corresponding sides.

Dividing land plots proportionally using parallel boundaries.

Operations such as scaling, rotation, reflection, and translation, which preserve similarity.

A transformation that enlarges or reduces a figure proportionally—basis of similarity.

BPT (Thales’), its converse, Similarity criteria, Area ratio theorem, Pythagoras theorem with converse.

Typically 4–8 marks in CBSE Class 10, including one theorem-based proof question.

Check for equal angles first; then verify proportional sides.

Yes, because all angles are \(60^\circ\) and sides are proportional.

No, only if the angles also match.

Three positive integers \(a, b, c\) satisfying \(a^2 + b^2 = c^2\), e.g., (3,4,5).

Allow quick checking of right triangles without calculation.

Yes, all corresponding angles remain equal.

No, areas change by the square of scale factor.

If \(\frac{a}{b} = \frac{c}{d}\), then \(\frac{a+c}{b+d}\) is also an intermediate proportional ratio.

The point lies between the endpoints of the segment.

Not necessarily; their shapes may differ.

The sides must satisfy Pythagorean equality.

Proofs, ratio-based calculations, similarity applications, height-distance applications, MCQs.

Identify corresponding sides \(\Rightarrow\) set ratios \(\Rightarrow\) cross-multiply \(\Rightarrow\) solve.

By forming two similar triangles using shadows, poles, or angles of elevation.

Two right triangles with one equal acute angle.

Use the equation \(\frac{side_1}{side_2} = \text{scale factor}\).

Using wrong pair of sides for ratio; ratio must correspond to the intersected sides.

Yes, it deals with division of sides in a triangle using parallel lines.

Show equality of sides/angles using SSS/SAS/ASA/RHS.

Maps are reduced versions (scaled diagrams) using constant ratios.

AAA ensures same shape but not same size, so not congruence.

Drawing additional lines to help prove similarity/congruence.

Cameras (lens projection), GPS, surveying instruments, and theodolites.

To determine distances and elevations indirectly.

Sometimes median-based constructions produce smaller similar triangles.

Yes, orientation or position does not affect similarity.

All polygons can be divided into triangles, making them the building blocks of geometry.

No, they must also share one acute angle.

Check quickly whether the square of the longest side equals the sum of squares of other two.

Because it supports trigonometry, coordinate geometry, mensuration, and real-world calculations.

TRIANGLES-Exercise 6.1

TRIGONOMETRIC FUNCTIONS-Exercise 3.2

TRIGONOMETRIC FUNCTIONS-Exercise 3.1

TRIANGLES-Exercise 6.1

Maths - Exercise

Solution:

Solution:

Solution:

Frequently Asked Questions

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TRIANGLES-Exercise 6.1

TRIGONOMETRIC FUNCTIONS-Exercise 3.2

TRIGONOMETRIC FUNCTIONS-Exercise 3.1

TRIANGLES-Exercise 6.1

Maths - Exercise

Solution:

Solution:

Solution:

Frequently Asked Questions

1 What is a triangle?

2 What is meant by congruence of triangles?

3 What are the main congruence rules studied in Class X?

4 What is similarity of triangles?

5 What are the similarity criteria for triangles?

6 What is the AA similarity criterion?

7 State the Basic Proportionality Theorem (BPT).

8 What is Thales’ Theorem?

9 State the converse of BPT.

10 What is Pythagoras Theorem?

11 What is the Converse of Pythagoras Theorem?

12 What is the area ratio theorem?

13 What is meant by corresponding parts of triangles?

14 How do we prove triangles similar using SSS?

15 What is the RHS congruence criterion?

16 What is the significance of similarity in real life?

17 What is a scale factor?

18 What is the relation between sides of similar triangles?

19 What is the relation between areas of similar triangles?

20 What is the relation between perimeters of similar triangles?

21 Can two triangles be similar but not congruent?

22 What does “proportion” mean in triangle similarity?

23 What is a mid-point theorem?

24 How is the mid-point theorem related to BPT?

25 What is the use of BPT in constructions?

26 How do you identify corresponding angles?

27 What does “equidistant” mean?

28 What is a right triangle?

29 What are the sides of a right triangle called?

30 Why is similarity important for trigonometry?

31 How is Pythagoras Theorem used in coordinate geometry?

32 Can similarity apply to non-triangular shapes?

33 What is a proportional divider?

34 What are the applications of Pythagoras Theorem?

35 How do you prove BPT using similarity?

36 What is a real-life example of BPT?

37 What is a transformation in triangles?

38 What is dilation?

39 What are exam-oriented theorems from this chapter?

40 How many questions usually appear from this chapter?

41 What is the fastest way to identify similar triangles?

42 Are equilateral triangles always similar?

43 Are isosceles triangles always similar?

44 What is a Pythagorean triplet?

45 How do Pythagorean triplets help in exams?

46 Does similarity preserve angles?

47 Does similarity preserve area?

48 State one property of proportional segments.

49 What is meant by ‘divides a side internally’?

50 Can two triangles with equal areas be similar?

51 What is a condition for the converse of Pythagoras Theorem?

52 What type of questions are commonly asked?

53 How to quickly solve similarity numerical problems?

54 How do you use triangle similarity in heights and distances?

55 What is the simplest example of similar triangles?

56 How to find missing sides using similarity?

57 What is the common mistake in BPT questions?

58 Is the BPT applicable only in triangles?

59 How to justify that two triangles are congruent?

60 How are similar triangles used in maps?

61 Why is AAA a similarity rule but not a congruence rule?

62 What is auxiliary construction in triangle proofs?

63 What devices use triangle similarity internally?

64 How do surveyors use similar triangles?

65 What is the relation between medians and similarity?

66 Can triangles be similar even if orientation differs?

67 What is the importance of triangles in geometry?

68 Can two right triangles be similar if they only share a right angle?

69 How to recognise a Pythagorean triangle in exam?

70 Why is triangle similarity foundational for Class 10 geometry?

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