Complete Solutions and Summary of Triangles – NCERT Class 10, Mathematics, Chapter 6 – Summary, Questions, Answers, Extra Questions
Comprehensive summary and explanation of Chapter 6 'Triangles', covering similarity and congruence of figures, criteria for similarity (AAA, AA, SSS, SAS), Thales' (Basic Proportionality) Theorem, properties of similar triangles, indirect measurement applications, and detailed proofs including a simple proof of the Pythagoras Theorem—paired with solved NCERT problems and extra questions for Class X Mathematics.
Updated: 8 months ago
Triangles
Chapter 6: Mathematics - Ultimate Study Guide | NCERT Class 10 Notes, Questions, Examples & Quiz 2025
Full Chapter Summary & Detailed Notes - Triangles Class 10 NCERT
Overview & Key Concepts
- Chapter Goal: Understand similarity of triangles, properties, theorems like BPT, criteria (AAA, AA, SSS, SAS). Exam Focus: Proofs, applications in heights/distances. 2025 Updates: Real-life examples enhanced. Fun Fact: Thales measured pyramids using similarity. Core Idea: Similar figures have same shape, not size. Real-World: Mountain heights, moon distance.
- Wider Scope: Geometry foundation, trigonometry links.
6.1 Introduction
- Recall triangles from earlier classes, congruence (same shape and size).
- Now, similar figures: same shape, not necessarily size. E.g., all circles similar, all squares similar.
- Apply to triangles: similarity principles for indirect measurements like Mount Everest height, moon distance.
- Activity: Guess how measured without tape? Using shadows, proportions.
- Chapter discusses similarity, Pythagoras proof via similarity.
6.2 Similar Figures
- All congruent figures similar, but converse not true.
- Circles with different radii similar, not congruent.
- Squares/equilateral triangles of different sides similar.
- Polygons similar if corresponding angles equal, sides proportional (scale factor).
- Photographs example: Enlargements keep angles, ratios of sides.
- Activity 1: Bulb shadow enlarges quadrilateral, similar shape.
- Corresponding vertices: A' ↔ A, etc.
- Transitivity: If A~B and B~C, A~C.
- Not similar if only angles or only sides match (square-rectangle, square-rhombus).
Exercise 6.1
- Fill blanks: Circles similar, squares similar, equilateral triangles similar, polygons if angles equal and sides proportional.
- Examples: Similar - two circles, two equilateral triangles; Non-similar - circle and square, triangle and quadrilateral.
- Quadrilaterals in fig. not similar (different angles/sides).
6.3 Similarity of Triangles
- Triangles similar if angles equal and sides proportional.
- Equiangular triangles: Angles equal.
- Thales: Ratio of corresponding sides in equiangular triangles same.
- Basic Proportionality Theorem (BPT/Thales Theorem): Line parallel to one side divides other two proportionally.
- Proof: Areas, equal bases between parallels.
- Activity 2: Points on arm, parallel line divides proportionally.
- Converse: If divides proportionally, parallel.
- Activity 3: Equal parts on arms, joining shows parallels.
- Example 1: DE||BC, prove AD/AB = AE/AC.
- Example 2: Trapezium AB||DC, EF||AB, prove AE/ED = BF/FC.
- Example 3: PS/SQ = PT/TR, ∠PST=∠PRQ, prove PQR isosceles.
Exercise 6.2
- DE||BC, find lengths.
- Points on sides, check parallel.
- LM||CB, LN||CD, prove AM/AB = AN/AD.
- DE||AC, DF||AE, prove BF/FE = BE/EC.
- DE||OQ, DF||OR, prove EF||QR.
- Points on OP,OQ,OR, AB||PQ, AC||PR, prove BC||QR.
- Prove mid-point theorems using BPT/converse.
- Trapezium diagonals intersect proportionally.
- Diagonals intersect, show trapezium.
6.4 Criteria for Similarity of Triangles
- AAA: Angles equal, sides proportional.
- AA: Two angles equal (third by sum).
- Activity 4: Equal angles, measure sides proportional.
- Proof: Construct equal sides, parallel, congruent, ratios.
- SSS: Sides proportional, angles equal.
- Activity 5: Proportional sides, measure angles equal.
- Proof: Construct smaller, parallel, congruent, ratios.
- SAS: Sides proportional including equal angle.
- Activity 6: Proportional sides with included angle, similar.
- Proof: Construct, parallel, congruent.
- Example 4: PQ||RS, prove ∆POQ ~ ∆SOR.
- Example 5: Find ∠P in figure.
- Example 6: OA·OB=OC·OD, prove ∠A=∠C, ∠B=∠D.
- Example 7: Girl shadow, find length.
- Example 8: Medians proportional, similar.
Exercise 6.3
- Pairs similar, criteria, symbolic.
- Angles in trapezium.
- Diagonals intersect, OA/OC=OB/OD.
- PQ||RS, ∠1=∠2, prove ∆PQS ~ ∆TQR.
- ∠P=∠RTS, prove ∆RPQ ~ ∆RTS.
- ∆ABE ≅ ∆ACD, prove ∆ADE ~ ∆ABC.
- Altitudes intersect, prove similarities.
- Parallelogram, BE intersects CD, prove ∆ABE ~ ∆CFB.
- Right triangles, prove similar, ratios.
- Bisectors, similar triangles.
- Isosceles, perpendiculars, prove ∆ABD ~ ∆ECF.
- Medians proportional, similar.
- ∠ADC=∠BAC, CA²=CB·CD.
- Medians proportional again.
- Pole shadow, tower height.
- Medians, prove AB/PQ = AD/PM.
6.5 Summary
- Similar figures: shape same, size may differ.
- Congruent are similar, not converse.
- Polygons: angles equal, sides proportional.
- BPT and converse.
- AAA/AA, SSS, SAS criteria.
Why This Guide Stands Out
Complete chapter coverage: Notes, examples, Q&A (all NCERT + extras), quiz. Student-centric, exam-ready for 2025. Free & ad-free.
Key Themes & Tips
- Similarity: Angles, proportions.
- Theorems: BPT key.
- Criteria: AAA/AA, SSS, SAS.
- Tip: Draw diagrams; check correspondences.
Exam Case Studies
Proofs using criteria; word problems on heights.
Project & Group Ideas
- Measure shadows for heights; model similarities.
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