CBSE Class 10 Mathematics: NCERT Solutions Class 10 Maths Chapter 6 Triangles (2025-26 Guide)

In CBSE Class 10 Mathematics, mastering NCERT Solutions Class 10 Maths Chapter 6 Triangles is essential for scoring top marks in the 2025-26 board examinations. Chapter 6 marks a pivotal transition from the intuitive congruence learned in Class 9 to the analytical realm of geometric similarity. Geometry carries significant weightage in the board exam 10 blueprint, and Triangles forms the bedrock of deductive proofs, geometric proportions, and trigonometry. This comprehensive guide breaks down every core theorem, similarity criterion, and high-frequency CBSE question with step-by-step mathematical rigor.

Key Concepts in Class 10 Triangles

To master CBSE Mathematics at the Class 10 level, understanding the conceptual definitions and distinctions between geometric congruence and geometric similarity is fundamental.

1. Congruence vs. Similarity of Figures

Two geometric figures are defined as:

• Congruent Figures: Shapes that have the same shape and the same size. All corresponding angles are equal, and all corresponding sides are equal in length (ratio 1:1).
• Similar Figures: Shapes that have the same shape, but not necessarily the same size. All corresponding angles are equal, and all corresponding sides are in the same proportion (ratio).

Note: All congruent figures are similar, but all similar figures need not be congruent. For example, two equilateral triangles of side 3 cm and 6 cm are similar (Δ₁ ∼ Δ₂) but not congruent.

2. Basic Proportionality Theorem (Thales' Theorem - Theorem 6.1)

The Basic Proportionality Theorem (BPT) is one of the most frequently tested theorems in CBSE Class 10 board exams.

Statement: If a line is drawn parallel to one side of a triangle to intersect the other two sides in distinct points, the other two sides are divided in the same ratio.

Step-by-Step Formal Proof:

1. Given: A triangle ΔABC in which a line parallel to side BC intersects sides AB and AC at points D and E respectively (i.e., DE ∥ BC).
2. To Prove: AD / DB = AE / EC
3. Construction: Join BE and CD. Draw DM ⊥ AC and EN ⊥ AB.
4. Proof:
   Recall that Area of a triangle = ½ × Base × Height.
   Area(ΔADE) = ½ × AD × EN
   Area(ΔBDE) = ½ × DB × EN
   Therefore, the ratio of their areas is:
   Area(ΔADE) / Area(ΔBDE) = (½ × AD × EN) / (½ × DB × EN) = AD / DB — (Equation 1)
   
   Similarly, taking AE and EC as bases with altitude DM:
   Area(ΔADE) = ½ × AE × DM
   Area(ΔCDE) = ½ × EC × DM
   Area(ΔADE) / Area(ΔCDE) = (½ × AE × DM) / (½ × EC × DM) = AE / EC — (Equation 2)
   
   Now, observe ΔBDE and ΔCDE. Both triangles lie on the same base DE and between the same parallel lines DE and BC.
   According to geometric properties: Area(ΔBDE) = Area(ΔCDE) — (Equation 3)
   
   From Equations (1), (2), and (3), the left-hand sides are equal. Hence:
   AD / DB = AE / EC. (Hence Proved)

3. Converse of Basic Proportionality Theorem (Theorem 6.2)

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

If in ΔABC, a line DE cuts AB at D and AC at E such that AD / DB = AE / EC, then DE ∥ BC.

4. Criteria for Similarity of Triangles

Two triangles ΔABC and ΔDEF are similar (ΔABC ∼ ΔDEF) if and only if their corresponding angles are equal and their corresponding sides are proportional. Under the revised NCERT curriculum, three primary criteria establish similarity without proving all six conditions:

• AAA (Angle-Angle-Angle) / AA Similarity Criterion: If in two triangles, two angles of one triangle are respectively equal to two angles of the other triangle, then the two triangles are similar (∠A = ∠D, ∠B = ∠E ⇒ ΔABC ∼ ΔDEF).
• SSS (Side-Side-Side) Similarity Criterion: If in two triangles, sides of one triangle are proportional to the sides of the other triangle (AB / DE = BC / EF = CA / FD), then their corresponding angles are equal and the triangles are similar.
• SAS (Side-Angle-Side) Similarity Criterion: If one angle of a triangle is equal to one angle of the other triangle (∠A = ∠D) and the sides including these angles are proportional (AB / DE = AC / DF), then the two triangles are similar.

5. Real-World Applications of Triangle Similarity

Similarity is not purely theoretical; it powers practical real-world measurements and modern technology:

• Indirect Measurement (Thales' Shadow Method): Ancient mathematicians and modern civil engineers calculate the heights of tall trees, towers, or pyramids by comparing the length of their shadows to a rod of known height under identical sunlight angle.
• Cartography and Map Scaling: Topographic maps and architectural blueprints use scale factors based on similarity ratios to accurately represent vast terrains and structures in miniature formats.
• Computer Graphics and Lens Optics: Camera perspective projection, focal length calculations, and 3D rendering algorithms utilize proportional triangles to project three-dimensional coordinates onto two-dimensional screens.

Important CBSE Questions with Answers

Below are essential solved questions aligned with the latest CBSE question bank and previous year board exam 10 papers.

Question 1 (BPT Application on Trapezium)

Problem: ABCD is a trapezium with AB ∥ DC. Its diagonals AC and BD intersect each other at point O. Show that AO / BO = CO / DO.

Solution:

1. Given: Trapezium ABCD with AB ∥ DC and diagonals AC and BD intersecting at O.
2. To Prove: AO / BO = CO / DO (or equivalently, AO / CO = BO / DO).
3. Construction: Through point O, draw a line segment OE ∥ AB meeting AD at E. Since AB ∥ DC, OE is also parallel to DC.
4. Step 1: In ΔADC, OE ∥ DC. By Basic Proportionality Theorem (BPT):
   AE / ED = AO / CO — (Equation 1)
5. Step 2: In ΔDAB, EO ∥ AB. By Basic Proportionality Theorem (BPT):
   DE / EA = DO / BO ⇒ Taking reciprocals: AE / ED = BO / DO — (Equation 2)
6. Step 3: Equating Equations (1) and (2):
   AO / CO = BO / DO
   Rearranging terms: AO / BO = CO / DO. (Hence Proved)

Question 2 (Algebraic BPT Equation)

Problem: In ΔABC, DE ∥ BC such that D lies on AB and E lies on AC. If AD = x, DB = x − 2, AE = x + 2, and EC = x − 1, determine the value of x.

Solution:

1. By Basic Proportionality Theorem, since DE ∥ BC:
   AD / DB = AE / EC
2. Substitute the given algebraic expressions:
   x / (x − 2) = (x + 2) / (x − 1)
3. Cross-multiply:
   x(x − 1) = (x + 2)(x − 2)
4. Expand both sides:
   x² − x = x² − 4
5. Subtract x² from both sides:
   −x = −4 ⇒ x = 4
6. Answer: The value of x is 4 (which gives valid positive side lengths: AD = 4 cm, DB = 2 cm, AE = 6 cm, EC = 3 cm).

Question 3 (Real-World Shadow Problem - AA Criterion)

Problem: A vertical pole of length 6 m casts a shadow 4 m long on the ground, and at the same time a tower casts a shadow 28 m long. Find the height of the tower.

Solution:

1. Let AB be the pole (AB = 6 m) and BC be its shadow (BC = 4 m). Let PQ be the tower (height = h) and QR be its shadow (QR = 28 m).
2. In ΔABC and ΔPQR:
   • ∠B = ∠Q = 90° (Poles and towers stand vertically to the ground)
   • ∠C = ∠R (Angular elevation of the Sun is equal at the same time)
3. By AA Similarity Criterion, ΔABC ∼ ΔPQR.
4. Since corresponding sides of similar triangles are proportional:
   AB / PQ = BC / QR
5. Substitute the known values:
   6 / h = 4 / 28
6 / h = 1 / 7 ⇒ h = 6 × 7 = 42 m
6. Answer: The height of the tower is 42 meters.

Question 4 (Proving Geometric Products via Similarity)

Problem: In ΔABC, D is a point on side BC such that ∠ADC = ∠BAC. Prove that CA² = CB × CD.

Solution:

1. Consider ΔABC and ΔDAC:
   • ∠BAC = ∠ADC (Given)
   • ∠ACB = ∠DCA = ∠C (Common angle to both triangles)
2. Therefore, by AA Similarity Criterion:
   ΔABC ∼ ΔDAC
3. Since corresponding sides of similar triangles are in proportion:
   CA / CD = CB / CA
4. Cross-multiplying yields:
   CA² = CB × CD. (Hence Proved)

How to Prepare for This Topic

Securing full marks in the Triangles chapter for the CBSE Class 10 Mathematics board exam requires focused practice and structured presentation:

1. Master Theorem Proofs with Proper Format: For the 5-mark theorem questions (specifically Thales' Theorem), always write the four formal sections: Given, To Prove, Construction, and Proof. Include clear diagrams with all vertices and altitudes labeled.
2. Maintain Strict Vertex Order: When writing triangle similarity statements (e.g., ΔABC ∼ ΔPQR), ensure the vertices correspond to equal angles. A mismatch in letter ordering will lead to incorrect side proportion ratios.
3. State Theorem Reasons in Brackets: Always justify every geometric step with reasons like [By BPT], [Converse of BPT], or [AA similarity criterion] to avoid losing step marks.
4. Practice Competency-Based Case Studies: CBSE 2025-26 sample papers emphasize application-based problems such as heights, mirrors, scale models, and shadow measurements.

Where to Practice More

To achieve a perfect score in CBSE Class 10 Mathematics, regular problem-solving with board-standard questions is vital. Explore Theorify QPTool to generate customized practice papers, access official CBSE question bank modules, and download previous years' solved question papers with comprehensive step-by-step marking schemes.