Class 10 Maths Chapter 6 – Triangles | Similarity, Theorems, NCERT Solutions & CBSE Questions (2025–26)

 Class 10 CBSE Maths – Chapter 6:Triangles

🔹 Key Concepts and Theorems

✅ Similar Figures

• Figures having the same shape but not necessarily the same size.

• All congruent figures are similar but not all similar figures are congruent.

✅ Similar Triangles

• Two triangles are similar if:

  1. Their corresponding angles are equal

  2. Their corresponding sides are in the same ratio (proportional)

Criteria for Similarity of Triangles:

1. AA (Angle-Angle) Criterion: Two triangles are similar if two angles of one triangle are respectively equal to two angles of another triangle.

2. SSS (Side-Side-Side) Criterion: If the sides of two triangles are in the same ratio, then the triangles are similar.

3. SAS (Side-Angle-Side) Criterion: If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are in the same ratio.

✅ Basic Proportionality Theorem (Thales Theorem)

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

✅ Converse of Basic Proportionality Theorem

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

✅ Pythagoras Theorem

Statement: In a right-angled triangle, the square of the hypotenuse is equal to the sum of the squares of the other two sides.
$AB^2 = AC^2 + BC^2$

✅ Converse of Pythagoras Theorem

If in a triangle, the square of one side is equal to the sum of squares of the other two sides, then the angle opposite to the first side is a right angle.

✅ Areas of Similar Triangles

If two triangles are similar, then the ratio of the areas of these triangles is equal to the square of the ratio of their corresponding sides.
$\frac{Area_1}{Area_2} = \left( \frac{a}{b} \right)^2$

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📘 NCERT Solved Examples (Highlights)

Example:

In triangle ABC, DE || BC and intersects AB and AC at D and E respectively. Prove:
$\frac{AD}{DB} = \frac{AE}{EC}$
✅ Apply Basic Proportionality Theorem

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📄 Exercise 6.1 – Similar Figures

Q1. Fill in the blanks:

• All circles are similar.

• All squares are similar.

• Two polygons of the same number of sides are similar if their corresponding angles are equal and corresponding sides are proportional.

Q2. Give two different examples of pair of similar figures:

• Two equilateral triangles

• Two rectangles with length-to-breadth ratio same

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📄 Exercise 6.2 – Similarity of Triangles (AA, SSS, SAS)

Q1. In triangle ABC and triangle DEF, $\angle A = \angle D, \angle B = \angle E$, then prove the triangles are similar.

✅ By AA similarity criterion

Q2. Sides of triangles are 5, 6, 7 and 10, 12, 14 respectively. Are they similar?

✅ Check ratio: $\frac{5}{10} = \frac{6}{12} = \frac{7}{14} = \frac{1}{2} \Rightarrow$ SSS similarity

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📄 Exercise 6.3 – Basic Proportionality Theorem (BPT)

Q1. In triangle XYZ, if AB || YZ and intersects XY and XZ at A and B, prove:

$\frac{XA}{AY} = \frac{XB}{BZ}$
✅ Use BPT

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📄 Exercise 6.4 – Areas of Similar Triangles

Q1. Two similar triangles have sides in the ratio 4:9. Find ratio of their areas.

$\left( \frac{4}{9} \right)^2 = \frac{16}{81}$

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📄 Exercise 6.5 – Pythagoras Theorem

Q1. In triangle ABC, right-angled at B, AB = 6 cm, BC = 8 cm. Find AC.

$AC^2 = AB^2 + BC^2 = 36 + 64 = 100 \Rightarrow AC = 10 \text{ cm}$

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🔁 Summary and Revision Notes

• All circles and all squares are similar

• Triangles are similar if they satisfy AA, SSS or SAS criteria

• BPT and Pythagoras Theorems are frequently applied in proofs

• Use area ratio = square of side ratio for similar triangles

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📘 CBSE Previous Year Questions – Triangles

✅ 1-Mark

(2020) In triangles ABC and DEF, $\angle A = \angle D$, $\angle B = \angle E$. Are the triangles similar?
📌 Yes, by AA similarity

✅ 2-Mark

(2021) Prove that if a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides those sides in the same ratio.
📌 Basic Proportionality Theorem

✅ 3-Mark

(2019) In triangle ABC, DE || BC, AD = 3 cm, DB = 6 cm. Find ratio $\frac{AE}{EC}$
📌 By BPT: $\frac{AD}{DB} = \frac{AE}{EC} = \frac{1}{2} \Rightarrow AE : EC = 1:2$

✅ 4-Mark

(2018) Prove Pythagoras Theorem using similarity.
📌 Construct perpendicular and use similar triangles to derive $hypotenus{e}^2=bas{e}^2+heigh{t}^2$