Triangles
Chapter 6 | Mathematics
Complete NCERT notes, similarity of triangles, criteria (AAA, SSS, SAS), Basic Proportionality Theorem (BPT), Pythagoras theorem, area ratio theorem, and key points for CBSE Class 10 Board Exam 2025-26.
📖 Introduction to Triangles
Two figures are said to be congruent if they have the same shape and size. Two figures are similar if they have the same shape but not necessarily the same size.
💡 All congruent figures are similar, but not all similar figures are congruent.
🔺 Similarity of Triangles
Two triangles are similar if:
- Their corresponding angles are equal.
- Their corresponding sides are in the same ratio (proportional).
📐 Criteria for Similarity of Triangles
1️⃣ AAA (or AA) Criterion
If two angles of one triangle are equal to two angles of another triangle, the triangles are similar.
If ∠A = ∠D and ∠B = ∠E → ΔABC ~ ΔDEF
2️⃣ SSS Criterion
If the corresponding sides of two triangles are proportional, the triangles are similar.
If AB/DE = BC/EF = CA/FD → ΔABC ~ ΔDEF
3️⃣ SAS Criterion
If one angle of a triangle is equal to one angle of another triangle and the sides including these angles are proportional, the triangles are similar.
If ∠A = ∠D and AB/DE = AC/DF → ΔABC ~ ΔDEF
⚡ Basic Proportionality Theorem (Thales Theorem)
📜 Statement
If a line is drawn parallel to one side of a triangle intersecting the other two sides, then it divides the two sides in the same ratio.
💡 Converse: If a line divides two sides of a triangle in the same ratio, 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 squares of the other two sides.
AC² = AB² + BC² (where ∠B = 90°)
🔄 Converse of Pythagoras Theorem
If the square of one side of a triangle equals the sum of squares of the other two sides, then the triangle is right-angled.
📊 Ratio of Areas of Similar Triangles
📝 NCERT Solved Examples
Example 1: BPT Application
In ΔABC, DE∥BC, AD=2cm, DB=3cm, AE=4cm. Find EC.
AD/DB = AE/EC → 2/3 = 4/EC → EC = 6cm
Example 2: Pythagoras Theorem
Find the length of diagonal of a rectangle of length 15cm and breadth 8cm.
Diagonal² = 15² + 8² = 225+64=289 → Diagonal = 17cm
Example 3: Similarity Criteria
In ΔPQR, ∠P=70°, ∠Q=50°. In ΔXYZ, ∠X=70°, ∠Y=60°. Are the triangles similar?
∠R=60°, ∠Z=50°. Thus ∠P=∠X, ∠Q=∠Z, ∠R=∠Y → ΔPQR ~ ΔXZY (AAA)
Example 4: Area Ratio
If ΔABC ~ ΔDEF, AB=4cm, DE=6cm, area of ΔABC=48cm². Find area of ΔDEF.
ar(ΔABC)/ar(ΔDEF) = (4/6)² = 16/36 = 4/9 → 48/ar = 4/9 → ar = 108 cm²
⭐ Key Points for Board Exam
- Similar triangles: Corresponding angles equal, sides proportional.
- BPT: If a line is parallel to one side, it divides the other two sides proportionally.
- Converse of BPT: If a line divides sides proportionally, it is parallel to the third side.
- Pythagoras theorem: Only for right-angled triangles.
- Area ratio: Ratio of areas = square of ratio of corresponding sides.
- SSS similarity: All three sides proportional.
- SAS similarity: Two sides proportional and included angle equal.
- AA similarity: Two angles equal is sufficient (since third automatically equal).