Triangles Class 10 Notes – CBSE Board Complete Guide

Triangles is one of the most important geometry chapters in CBSE Class 10 Maths. This chapter covers similarity of triangles, criteria for similarity, Basic Proportionality Theorem (Thales Theorem), Pythagoras Theorem and its converse. Questions from this chapter frequently appear in board exams for 6–8 marks. These detailed notes are prepared for conceptual clarity, exam confidence and full-score preparation.

1. Similar Figures and Similar Triangles

Two figures are said to be similar if they have the same shape but not necessarily the same size. In triangles, similarity means:

If triangle ABC is similar to triangle DEF, we write: △ABC ~ △DEF

Similarity plays a crucial role in solving height-distance problems, area ratio questions and proof-based board questions.

2. Criteria for Similarity of Triangles

(i) AAA Similarity Criterion

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

This criterion is widely used in proof-based questions.

(ii) SAS Similarity Criterion

If two sides of one triangle are proportional to two sides of another triangle and the included angle is equal, then the triangles are similar.

(iii) SSS Similarity Criterion

If all three sides of one triangle are proportional to the corresponding sides of another triangle, then the triangles are similar.

3. Basic Proportionality Theorem (Thales Theorem)

If a line is drawn parallel to one side of a triangle to intersect the other two sides, then it divides those sides in the same ratio.

This theorem is extremely important for 3–4 mark proof questions.

4. Converse of Basic Proportionality Theorem

If a line divides two sides of a triangle in the same ratio, then the line is parallel to the third side.

This is frequently asked in board exams for proof.

5. Pythagoras Theorem

In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.

If triangle ABC is right-angled at B: AC² = AB² + BC²

6. Converse of Pythagoras Theorem

If in a triangle, square of one side equals the sum of squares of the other two sides, then the triangle is right-angled.

7. Areas of Similar Triangles

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

This formula is commonly asked in 2–3 mark questions.

8. Solved Examples (Board Pattern)

Example 1

Check whether triangles with sides 3 cm, 4 cm, 5 cm form a right triangle.

Solution: 5² = 25 3² + 4² = 9 + 16 = 25 Since both are equal, triangle is right-angled.

Example 2

The sides of two similar triangles are in the ratio 4:7. Find ratio of their areas.

Area ratio = (4/7)² = 16/49

Example 3

In △ABC, DE || BC. If AD = 3 cm, DB = 2 cm, AE = 6 cm, find EC.

Using BPT: AD / DB = AE / EC 3 / 2 = 6 / EC EC = 4 cm

9. Previous Year Board Questions (PYQ)

1. Prove that if a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
2. Prove Pythagoras theorem using similarity.
3. Find ratio of areas of two similar triangles if corresponding sides are 5 cm and 8 cm.
4. Check whether given triangle is right-angled.
5. Use similarity to find height of a building using shadow method.

10. Important Case Study Questions

1. A tower casts a shadow of 20 m when a stick of 1 m casts a shadow of 2 m. Find height of tower.
2. A ladder leaning against a wall forms a right triangle. Verify using Pythagoras theorem.
3. Two triangular parks have sides in ratio 3:5. Compare their areas.

11. Exam Preparation Strategy

To score full marks in Triangles:

• Memorise all similarity criteria.
• Practice at least 5 proof-based questions.
• Understand area ratio concept clearly.
• Revise Pythagoras theorem daily.
• Solve previous 10 years CBSE questions.

12. Common Mistakes to Avoid

• Forgetting to write similarity symbol (~).
• Not writing reason in proof questions.
• Mixing up side ratios in SSS similarity.
• Forgetting to square ratio in area problems.

Conclusion

Triangles is a high-scoring and concept-based chapter in Class 10 Maths. With strong understanding of similarity criteria, BPT and Pythagoras theorem, students can easily score full marks in this chapter. Regular practice of board-level questions ensures confidence and accuracy in exams.

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