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I. Chapter Summary

This chapter explores the properties of triangles, focusing on congruence and the criteria used to prove triangle congruency. It also introduces important theorems, such as the angle sum property, properties of isosceles triangles, and inequalities in triangles. Students learn how to apply congruence rules to solve geometrical problems logically. This chapter is a key stepping stone for advanced geometry in higher classes.

II. Key Concepts Covered

Concept	Explanation
Triangle	A closed figure with three sides, three angles, and three vertices.
Congruent Triangles	Two triangles are congruent if they have exactly the same size and shape.
Congruence Criteria
→ SSS (Side-Side-Side)
→ SAS (Side-Angle-Side)
→ ASA (Angle-Side-Angle)
→ AAS (Angle-Angle-Side)
→ RHS (Right angle-Hypotenuse-Side, only for right-angled triangles)
Angle Sum Property	Sum of the interior angles of a triangle is always 180°.
Properties of Isosceles Triangle	Angles opposite equal sides are equal.
Triangle Inequality Theorem	The sum of the lengths of any two sides of a triangle is greater than the third side.
Exterior Angle Theorem	An exterior angle of a triangle is equal to the sum of the two interior opposite angles.

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. Which of the following is not a congruence criterion?
   a) SSS
   b) ASA
   c) AAA ✔️
   d) SAS
2. The sum of all angles in a triangle is:
   a) 180° ✔️
   b) 90°
   c) 270°
   d) 360°
3. Two triangles are congruent if they have:
   a) Equal areas
   b) Equal perimeters
   c) Equal angles only
   d) Same shape and same size ✔️
4. If two sides of a triangle are 5 cm and 6 cm, then the third side must be:
   a) 12 cm
   b) more than 1 cm
   c) less than 11 cm ✔️
   d) equal to the sum of both sides

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

1. Using ASA criterion, prove two triangles are congruent if two angles and the included side are equal.
2. In triangle ABC, AB = AC and ∠B = 50°. Find ∠C. (PYQ 2020)
3. State and prove the angle sum property of a triangle using a figure.
4. In triangle XYZ, XY = XZ. Prove that ∠Y = ∠Z. (PYQ 2019)

(C) Long Answer Questions (5 Marks)

1. Prove the Triangle Inequality Theorem and explain its applications.
2. In triangle ABC and triangle PQR, it is given that AB = PQ, BC = QR, and ∠B = ∠Q. Prove the triangles are congruent.
3. An exterior angle of a triangle is 110° and one interior opposite angle is 40°. Find the third angle and the remaining interior angle. Show your working with a diagram. (PYQ 2018)
4. Construct a triangle given two angles and a side between them. Prove it congruent to another triangle with the same data using ASA criterion.

(D) HOTS (Higher Order Thinking Skills)

1. Can a triangle have two right angles? Justify your answer using the angle sum property.
2. If two triangles have equal corresponding angles, are they necessarily congruent? Why or why not? Give an example.

IV. Key Formulas/Concepts

Property	Statement / Formula
Angle Sum Property	∠A + ∠B + ∠C = 180°
Exterior Angle Theorem	Exterior ∠ = ∠Interior 1 + ∠Interior 2
Congruence Criteria	SSS, SAS, ASA, AAS, RHS
Isosceles Triangle Property	Angles opposite equal sides are equal
Triangle Inequality	Sum of any two sides > third side

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)

Chapter	Estimated Marks	Type of Questions Typically Asked
Triangles	6–8 Marks	MCQ, Short Proofs, Diagram-Based Long Questions, HOTS

VII. Previous Year Questions (PYQs)

Marks	Question	Year
1 mark	State the sum of angles of a triangle.	PYQ 2020
2 marks	If AB = AC and ∠B = 50°, find ∠C.	PYQ 2020
3 marks	Prove that angles opposite to equal sides are equal.	PYQ 2019
5 marks	Prove the triangle inequality theorem with diagram.	PYQ 2018

VIII. Real-World Application Examples

• Architecture: Triangular trusses for strength in bridges and buildings.
• Robotics: Triangular components ensure rigidity in mechanical arms.
• Surveying: Land is divided and measured using triangle-based calculations.
• Art and Design: Symmetrical triangular patterns in mandalas and mosaics.

IX. Student Tips & Strategies for Success

 Time Management

• Day 1: Congruence criteria & properties
• Day 2: Angle sum, exterior angle, isosceles triangle
• Day 3: Triangle inequality & construction-based questions

 Exam Preparation

• Practice proofs with diagrams
• Memorize congruence criteria with real examples
• Write step-by-step logical reasoning in answers

 Stress Management

• Use flashcards for criteria & theorems
• Teach a friend one congruence proof—it boosts clarity

X. Career Guidance & Exploration

• For Classes 9–10:
  ➤ Foundation for logical proofs and constructions
  ➤ Useful for NTSE, Olympiads, and SOF IMO
• For Classes 11–12 & Beyond:
  ➤ Strong base for:
  • Civil Engineering
  • Architecture
  • Mathematics
  • Animation & Game Design
  • Aviation (navigation & radar triangulation)
• Entrance Exams:
  ➤ JEE (Geometry Section)
  ➤ NDA (Visual-spatial & mathematical reasoning)
  ➤ CUET
  ➤ RMO & INMO

XI. Important Notes

 Focus on drawing neat labeled diagrams for all proofs.
 Clearly differentiate between congruence criteria.
 Always include reasons/justifications for each step in a proof.
 Practice real-life triangle questions to enhance practical understanding.