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I. Chapter Summary

This chapter explores the fundamental concepts of similarity of triangles, which builds upon the knowledge of congruence from earlier classes. Students learn about the criteria for similarity, properties of similar triangles, and how similarity leads to proportionality in sides and equality in angles. It also covers the Pythagoras Theorem and its converse, which are crucial tools in geometry and trigonometry applications.

II. Key Concepts Covered

Concept	Explanation
Similar Figures	Figures having the same shape but not necessarily the same size.
Similarity of Triangles	Two triangles are similar if their corresponding angles are equal and corresponding sides are in the same ratio.
Criteria for Similarity
1. A (Angle)
2. S (Side)
3. SAS (Side-Angle-Side)
Basic Proportionality Theorem (Thales’ Theorem)	If a line is drawn parallel to one side of a triangle, it divides the other two sides in the same ratio.
Pythagoras Theorem	In a right-angled triangle:
$a^2 + b^2 = c^2$
Converse of Pythagoras Theorem	If the square of one side of a triangle is equal to the sum of the squares of the other two sides, the triangle is right-angled.

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. If two triangles are similar, then their corresponding angles are:
   a) Equal ✅ (PYQ 2021)
   b) Proportional
   c) Supplementary
   d) Complementary
2. In triangle ABC, DE ∥ BC and AD = 3 cm, DB = 2 cm. Find AE/EC.
   a) $frac{3}{2}$✅
   b) $frac{2}{3}$
   c) 1
   d) None
3. In a triangle, if one angle is 90°, and the sides satisfy $a^2 + b^2 = c^2$, then the triangle is:
   a) Scalene
   b) Right-angled ✅
   c) Equilateral
   d) Acute
4. If two triangles are similar, then the ratio of their areas is equal to the square of the ratio of their:
   a) Perimeters
   b) Sides ✅
   c) Angles
   d) Heights

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

1. In triangle ABC, D and E are points on AB and AC respectively such that DE ∥ BC. Prove: $frac{AD}{DB} = frac{AE}{EC}​$.
2. If two triangles are similar and the ratio of their sides is 2:3, find the ratio of their areas.
3. Using Pythagoras Theorem, find the length of the hypotenuse if the other two sides are 6 cm and 8 cm.
4. In triangle PQR, a line parallel to QR intersects PR and PQ in points A and B respectively. Prove that triangle ABP ∼ triangle QRP.

(C) Long Answer Questions (5 Marks)

1. In triangle ABC, D is a point on AB and E is a point on AC such that $DE parallel BC$. Prove that $frac{AD}{DB} = frac{AE}{EC}$ and $angle ADE = angle ABC, quad angle DEA = angle ACB$
   
2. Prove Pythagoras Theorem using similarity.
3. In two similar triangles ABC and DEF, if AB = 3 cm, BC = 4 cm, CA = 5 cm and DE = 6 cm, find lengths of EF and FD.
4. A tree 6 m high casts a shadow 4 m long. At the same time, a pole casts a shadow 10 m long. Find the height of the pole.

(D) HOTS (Higher Order Thinking Skills)

1. Two poles of heights 6 m and 3 m are standing opposite each other on either side of a road, which is 30 m wide. Find the point on the road where the line joining the tops of the poles meets the ground.
2. In triangle XYZ, a line DE is drawn such that $frac{DX}{XY} = frac{EZ}{ZY}$. Prove that $DE parallel XZ$
   .

IV. Key Formulas/Concepts

Formula/Concept	Description
Basic Proportionality Theorem	$text{If } DE parallel BC text{ in } triangle ABC, text{ and } D in AB, E in AC, text{ then:} \ frac{AD}{DB} = frac{AE}{EC}$
Pythagoras Theorem	$a^2 + b^2 = c^2$
Ratio of Areas	$frac{text{Area}_1}{text{Area}_2} = left( frac{text{Side}_1}{text{Side}_2} right)^2$
Similar Triangles	$text{If } triangle ABC sim triangle DEF, text{ then:} \ frac{AB}{DE} = frac{BC}{EF} = frac{CA}{FD}$

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
Chapter 6: Triangles	6–8 Marks	1 Long Answer, 2 Short Answers, 1 MCQ

VII. Previous Year Questions (PYQs)

Year	Marks	Question
2021	1	Similarity implies corresponding angles are equal
2020	3	Prove Basic Proportionality Theorem
2019	5	Use similarity to prove Pythagoras Theorem
2018	2	Find side using triangle similarity

VIII. Real-World Application Examples

Context	Application
Architecture	Designing similar shapes in different scales
Maps & Models	Scale drawings rely on triangle similarity
Surveying	Triangulation used in land measurement
Navigation	GPS triangulation uses Pythagorean relationships

IX. Student Tips & Strategies for Success

Time Management

• Dedicate time to draw accurate diagrams.
• Practice labeling points and writing correct ratios.

Exam Preparation

• Learn proofs of BPT and Pythagoras Theorem.
• Solve problems involving real-life applications.

Stress Management

• Break down theorems into steps and practice writing them from memory.
• Use triangle similarity in word problems for better clarity.

X. Career Guidance & Exploration (Class-Specific)

For Classes 9–10

Streams

• Science: Architecture, Engineering
• Commerce: Business Modeling, Technical Drawing
• Arts: Design, Geography

Careers

• Architect, Civil Engineer, Cartographer

Exams

• NTSE, RMO, CUET, Olympiads

XI. Important Notes

• Refer only to official CBSE/NCERT websites for updates.
• Draw proper diagrams—this often earns full marks even if steps are partially correct.
• Focus more on understanding triangle properties than memorizing figures.