Class 10 Mathematics Chapter 6 Triangles

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Class 10 Mathematics -> Chapter 6: Triangles

I. Chapter Summary:

This chapter focuses on the properties of triangles and the different criteria used to prove their congruence and similarity. Students will explore concepts such as the Pythagorean Theorem, congruence of triangles, and similarity of triangles, along with the criteria for congruence like SSS, SAS, ASA, and RHS. The chapter also emphasizes the importance of proportionality and the use of similarity in solving practical problems. By the end of the chapter, students will be able to apply these properties to real-life situations and mathematical problems.

II. Key Concepts Covered:

  1. Basic Properties of Triangles:
    • A triangle is a polygon with three sides and three angles.
    • The sum of the interior angles of any triangle is always 180°.
  2. Congruence of Triangles:
    • Congruence means that two triangles are identical in shape and size, though they may be rotated or flipped.
    • Criteria for triangle congruence:
      • SSS (Side-Side-Side): If three sides of one triangle are equal to three sides of another triangle, then the triangles are congruent.
      • SAS (Side-Angle-Side): If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
      • ASA (Angle-Side-Angle): If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
      • RHS (Right Angle-Hypotenuse-Side): 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.
  3. Similarity of Triangles:
    • Similarity means that two triangles have the same shape but may differ in size.
    • Criteria for triangle similarity:
      • AA (Angle-Angle): If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
      • SSS (Side-Side-Side): If the corresponding sides of two triangles are proportional, the triangles are similar.
      • SAS (Side-Angle-Side): If one angle of a triangle is equal to one angle of another triangle and the sides containing these angles are in the same ratio, the triangles are similar.
  4. Pythagoras Theorem:

    • In a right-angled triangle, the square of the length of the hypotenuse is equal to the sum of the squares of the lengths of the other two sides:

      $c^2 = a^2 + b^2$

      Where c is the hypotenuse and a and b are the other two sides.

  5. Applications of Similar Triangles:
    • Proportionality Theorem: If a line is drawn parallel to one side of a triangle, it divides the other two sides proportionally.
    • Area of Similar Triangles: The areas of two similar triangles are in the square of the ratio of their corresponding sides.

III. Important Questions:

(A) Multiple Choice Questions (MCQs) (1 Mark):

  1. In the triangles ABC and DEF, if $angle A = angle D, quad AB = DE, quad AC = DF$ then the triangles are:
    • (A) Similar by AA criterion
    • (B) Congruent by ASA criterion
    • (C) Congruent by SSS criterion
    • (D) Similar by SAS criterion
    • Answer: (B)
  2. Which of the following is not a criterion for the congruence of triangles?
    • (A) SSS
    • (B) SAS
    • (C) SSA
    • (D) ASA
    • Answer: (C)
  3. The sum of the angles in a triangle is:
    • (A) 90°
    • (B) 180°
    • (C) 270°
    • (D) 360°
    • Answer: (B)
  4. In a right-angled triangle, the sides 9 cm, 12 cm, and 15 cm form a:
    • (A) Scalene triangle
    • (B) Isosceles triangle
    • (C) Equilateral triangle
    • (D) Right-angled triangle
    • Answer: (D)

(B) Short Answer Questions (2/3 Marks):

  1. Prove that the sum of the angles in a triangle is 180°.
  2. In an equilateral triangle, if the length of each side is 6 cm, find the area of the triangle.
  3. Using the Pythagoras Theorem, verify whether the sides 7 cm, 24 cm, and 25 cm form a right-angled triangle.
  4. In two similar triangles, the ratio of the corresponding sides is 3:5. Find the ratio of their areas.

(C) Long Answer Questions (5 Marks):

  1. Prove that if two triangles are congruent, then their corresponding parts are equal.
  2. Two triangles ABC and PQR are such that $angle A = angle P, quad AB = 3 times PQ, quad BC = 3 times QR$. Prove that the triangles are similar.
  3. In the figure, a line parallel to one side of a triangle intersects the other two sides. Prove that the line divides the two sides proportionally.
  4. Using the Pythagorean Theorem, find the length of the hypotenuse of a right-angled triangle whose legs are 8 cm and 15 cm long.

(D) HOTS (Higher Order Thinking Skills) Questions:

  1. Two triangles have their corresponding sides in the ratio 2:3. If the area of the smaller triangle is 48 cm², find the area of the larger triangle.
  2. A tower casts a shadow of 30 m on the ground. If the angle of elevation of the sun is 30°, find the height of the tower using trigonometric ratios.

IV. Key Formulas/Concepts:

  1. Congruence of Triangles:
    • SSS: If three sides of one triangle are equal to the corresponding three sides of another triangle, the triangles are congruent.
    • SAS: If two sides and the included angle of one triangle are equal to two sides and the included angle of another triangle, the triangles are congruent.
    • ASA: If two angles and the included side of one triangle are equal to two angles and the included side of another triangle, the triangles are congruent.
    • RHS: In right-angled triangles, if the hypotenuse and one side are equal, the triangles are congruent.
  2. Similarity of Triangles:
    • AA: If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
    • SSS: If the corresponding sides of two triangles are proportional, the triangles are similar.
    • SAS: If one angle is equal and the sides containing the angle are in the same ratio, the triangles are similar.
  3. Pythagorean Theorem:
    • $c^2 = a^2 + b^2$
      Where ccc is the hypotenuse and a and b are the legs of the triangle.

V. Deleted Portions (CBSE 2025-2026):

No portions have been deleted from this chapter as per the rationalized NCERT textbooks.

VI. Chapter-Wise Marks Bifurcation (Estimated – CBSE 2025-2026):

Unit/Chapter Estimated Marks Type of Questions Typically Asked
Triangles 6 1 Mark (MCQs), 2/3 Marks (Short Answers), 5 Marks (Long Answers)

VII. Previous Year Questions (PYQs):

  1. 2019:
    • (1 Mark) If two triangles have corresponding angles equal, what can be concluded about the triangles?
    • (2 Marks) Show that in a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
    • (5 Marks) In the figure, if $triangle ABC sim triangle PQR$, prove that the ratio of their areas is the square of the ratio of their corresponding sides.
  2. 2020:
    • (3 Marks) Prove that the sum of the angles of a triangle is 180°.
    • (5 Marks) In two triangles, the corresponding sides are proportional. Prove that they are similar.

VIII. Real-World Application Examples to Connect with Topics:

  1. Architecture and Construction:
    • Triangular structures are used in buildings for stability. Congruent triangles are used to design roofs, bridges, and towers.
  2. Navigation and Surveying:
    • Triangular calculations, including similarity and congruence, help in determining distances, heights, and locations in navigation and surveying.

IX. Student Tips & Strategies for Success:

  1. Time Management:
    • Focus on understanding the basic properties first, then practice applying the formulas in real-world scenarios.
  2. Exam Preparation:
    • Work on solving past year questions. Try to apply the congruence and similarity criteria systematically to gain a deeper understanding.
  3. Stress Management:
    • Stay calm and organize your study schedule. Practice relaxation techniques like deep breathing during study breaks.

X. Career Guidance & Exploration (Class-Specific):

  • For Classes 9-10:
    • A strong foundation in triangles will help you pursue careers in Engineering (especially Civil, Mechanical, and Architecture), Mathematics, Physics, and Designing.
  • For Classes 11-12:
    • Triangular concepts are foundational for JEE, NEET, NDA, and AIIMS exams. Careers in Architecture, Urban Planning, Surveying, and Robotics are linked to understanding triangles.

XI. Important Notes:

  • Regular revision and practice are crucial for mastering triangle properties and formulas.
  • Refer to the official CBSE website and NCERT textbooks for the most updated syllabus and exam guidelines.

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