NCERT Solutions Class 10 Maths Triangles Exercise 6.1

Welcome to the beginning of your journey with Triangles in CBSE Class 10! Exercise 6.1 serves as the foundational stepping stone for Chapter 6. While Class 9 focused deeply on the 'congruency' of triangles, Class 10 introduces you to the fascinating world of 'similarity'. Understanding similarity is not just essential for scoring high in your CBSE Board exams, but it also forms the basis of real-world concepts like map-scaling, architectural blue-printing, and 3D modeling. In this guide, you will master the fundamental difference between congruent and similar shapes. You will learn the specific conditions that make geometric polygons similar, walk through step-by-step solutions for triangles ex 6 1 class 10 ncert, and test your skills with carefully designed practice questions. Let's make triangles simple and fun with YoLearn AI!

Understanding Similarity: The Core Concept

To master CBSE Class 10 Triangles, you must first grasp the core concept of Similarity. In geometry, two shapes are said to be similar if they have the exact same shape, but not necessarily the same size. Think of a photograph and its enlargement: everything looks identical in proportion, but one is bigger than the other.

Historically, you studied congruent figures in Class 9. Two figures are congruent if they have both the same shape and the same size. Hence, all congruent figures are similar, but similar figures do not need to be congruent.

For any two polygons with the same number of sides to be similar, they must satisfy two essential conditions simultaneously:

  1. Their corresponding angles must be equal.
  2. Their corresponding sides must be in the same ratio (proportional).

If even one of these conditions is violated, the polygons are not similar. For instance, a square and a rectangle have equal corresponding angles (all 90 degrees), but their sides are not in the same ratio. Thus, they are not similar.

Important Definitions

Congruent Figures
Geometric figures that have identical shapes and identical sizes.
Similar Figures
Geometric figures that have the same shape but different sizes.
Scale Factor
The constant ratio of the corresponding sides of similar polygons.
Equilateral Triangles
Triangles with all three sides equal and all three angles equal to 60 degrees. All equilateral triangles are always similar.

How to Determine if Two Figures are Similar

  1. Check the Number of Sides — Ensure both polygons have the exact same number of sides. You cannot compare a triangle with a quadrilateral for similarity.
  2. Compare Corresponding Angles — Verify if every interior angle in the first polygon has an equal counterpart in the second polygon. Angle measures must match exactly.
  3. Calculate the Ratio of Sides — Divide the lengths of corresponding sides. If all ratios are equal (e.g., AB/PQ = BC/QR = CD/RS = DA/SP), the sides are proportional.
  4. Conclude similarity — If both Step 2 (equal angles) and Step 3 (proportional sides) are met, the figures are similar. If either fails, they are non-similar.

Common Mistakes to Avoid in Exercise 6.1

1. Confusing 'Similar' with 'Congruent': Remember that all congruent circles are similar, but not all similar circles are congruent. A circle of radius 2 cm and a circle of radius 5 cm are similar, but not congruent.

2. Checking only one condition for similarity: Students often see that two quadrilaterals have equal corresponding angles and assume they are similar. Remember, both equal angles and proportional sides must be satisfied. For example, a square and a rectangle both have 90-degree angles, but their sides are not proportional, so they are not similar.

3. Incorrect order of vertices: When writing similarity statements (e.g., Triangle ABC is similar to Triangle PQR), the order of vertices must represent the corresponding angles.

Practice Questions with Solutions

  • Q: Fill in the blanks with the correct words: (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) A: Step 1: Analyze each fill-in-the-blank question based on the geometric properties of similarity. Step 2: (i) All circles are similar because they have the same shape but different radii. Step 3: (ii) All squares are similar because their angles are always 90° and their sides are always in equal ratios. Step 4: (iii) All equilateral triangles are similar as their angles are always 60° and sides are proportional. Step 5: (iv) Two polygons are similar if their corresponding angles are equal and their corresponding sides are proportional. Final answer: (i) similar, (ii) similar, (iii) equilateral, (iv) equal, proportional.
  • Q: Give two different examples of a pair of non-similar figures. A: Step 1: Identify pairs of shapes where shapes either have different geometries or fail similarity rules. Step 2: Example 1: A circle and a triangle. These are completely different shapes and cannot be mapped to each other. Step 3: Example 2: A rectangle and a rhombus. Though both are quadrilaterals, their corresponding angles are not equal (a rectangle has 90° angles, whereas a rhombus does not). Final answer: Pair 1: A circle and a triangle; Pair 2: A rectangle and a rhombus.
  • Q: State whether the following quadrilaterals are similar or not: Quadrilateral PQRS has sides of 1.5 cm and angles that are not 90 degrees (it is a rhombus). Quadrilateral ABCD is a square with sides of 3 cm. A: Step 1: Check the first condition of similarity: Corresponding angles must be equal. The first figure is a rhombus (angles are not 90°), and the second is a square (all angles are 90°). Thus, corresponding angles are not equal. Step 2: Check the second condition of similarity: Corresponding sides must be proportional. Ratio of sides = 1.5 / 3 = 1/2. The sides are proportional, but the first condition (angles) is violated. Step 3: Since both conditions are not satisfied, they are not similar. Final answer: The quadrilaterals are not similar because their corresponding angles are not equal.
  • Q: Why are all congruent triangles also similar, but all similar triangles are not necessarily congruent? Give a mathematical reason. A: Step 1: Let two triangles be congruent. This means their corresponding angles are equal, and their corresponding sides are equal. Step 2: Since corresponding sides are equal, their ratio is 1:1. Hence, the sides are proportional. This satisfies both conditions of similarity, meaning congruent triangles are always similar. Step 3: Now let two triangles be similar with a scale factor of 1:2. Their angles are equal, but their sides are not equal (one is twice as large). Hence, they are not congruent. Final answer: Congruent triangles satisfy similarity conditions with a ratio of 1, but similar triangles can have unequal side lengths, making them non-congruent.

Frequently Asked Questions

What is the main difference between similarity and congruency?

Congruent figures have both the same shape and the same size. Similar figures have the same shape, but their sizes can be different because their sides are scaled proportionally.

Are all isosceles triangles similar?

No, all isosceles triangles are not similar. For similarity, their corresponding angles must be equal, which is not always true for all isosceles triangles. Only equilateral triangles are guaranteed to be similar.

Why are a square and a rhombus of different sizes not similar?

While their corresponding sides might be proportional, their corresponding angles are not equal (a square has all 90-degree angles, while a rhombus does not). Therefore, they fail the angle equality condition of similarity.