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I. Chapter Summary

This chapter introduces the basic concepts of lines and angles, laying the foundation for geometrical reasoning. It covers intersecting and parallel lines, types of angles, and relationships between angles formed when a transversal cuts parallel lines. The chapter also includes important angle properties such as linear pair, vertically opposite angles, and angle sum property of a triangle. Students learn to apply logical steps to prove basic geometrical results.

II. Key Concepts Covered

Concept	Explanation
Line and Ray	A line extends indefinitely in both directions; a ray has a fixed starting point and extends in one direction.
Angle	Formed when two rays meet at a common point (vertex). Measured in degrees (°).
Types of Angles
– Acute $< 90^circ$
– Right $= 90^circ$
– Obtuse $> 90^circ text{ and } < 180^circ$
– Straight $= 180^circ$
– Reflex $> 180^circ$
Linear Pair	Two adjacent angles whose non-common arms form a straight line. Their sum = 180°.
Vertically Opposite Angles	When two lines intersect, opposite angles are equal.
Transversal	A line intersecting two or more lines at different points.
Angles formed by Transversal
– Corresponding angles
– Alternate interior angles
– Alternate exterior angles
– Co-interior (consecutive interior) angles
Angle Sum Property of Triangle	Sum of all interior angles of a triangle = $180^circ$ .

III. Important Questions

(A) Multiple Choice Questions (1 Mark)

1. If two angles form a linear pair, then they are:
   a) Complementary
   b) Supplementary ✔️
   c) Equal
   d) Right angles
2. Vertically opposite angles are always:
   a) Supplementary
   b) Complementary
   c) Equal ✔️
   d) Unequal
3. The sum of the angles of a triangle is always:
   a) 90°
   b) 270°
   c) 360°
   d) 180° ✔️
4. If one angle of a linear pair is 70°, the other is:
   a) 90°
   b) 70°
   c) 110° ✔️
   d) 180°

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

1. In the figure, lines AB and CD intersect at O. If $angle AOC = 40^circ$
   , find $angle BOD$.
2. Find the measure of each angle if two complementary angles differ by 30°.
3. If a transversal intersects two parallel lines and one pair of alternate interior angles are 65°, find the other angles.
4. Using a diagram, explain the angle sum property of a triangle.

(C) Long Answer Questions (5 Marks)

1. Prove that if two lines intersect, the vertically opposite angles are equal.
2. In a triangle, one angle is twice the second and the third angle is 40° more than the second. Find all the angles.
3. Draw a figure where a transversal cuts two lines. Identify all pairs of corresponding, alternate interior, and co-interior angles.
4. Prove that the sum of the angles of a triangle is 180° using the concept of a parallel line and transversal.

(D) HOTS (Higher Order Thinking Skills)

1. Can two acute angles form a linear pair? Why or why not?
2. A straight line is drawn from a point inside a triangle such that it forms two equal angles with two sides. What type of triangle can it be? Explain.

IV. Key Formulas/Concepts

Concept	Formula/Property
Linear Pair	$angle A + angle B = 180^circ$
Vertically Opposite Angles	$angle 1 = angle 3, quad angle 2 = angle 4$
Complementary Angles	$angle A + angle B = 90^circ$
Supplementary Angles	$angle A + angle B = 180^circ$
Angle Sum of Triangle	$angle A + angle B + angle C = 180^circ$
Transversal Angle Relations
$- angle text{Corresponding} = text{Equal}$
$- angle text{Alternate interior} = text{Equal}$
$- angle text{Co-interior} = text{Supplementary}$

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	Types of Questions Typically Asked
Lines and Angles	6–8 Marks	1 MCQ, 1 Short Answer, 1 Long Answer (proof/diagram), 1 HOTS

VII. Previous Year Questions (PYQs)

Marks	Question	Year
1 mark	Vertically opposite angles are always…	PYQ 2020
2 marks	Two complementary angles differ by 10°. Find the angles.	PYQ 2019
3 marks	Find the values of unknown angles when two parallel lines are cut by a transversal.	PYQ 2020
5 marks	Prove angle sum property of triangle with a diagram.	PYQ 2018

VIII. Real-World Application Examples

• Architecture: Angles between beams and supports.
• Navigation: Triangle and angle calculations in GPS and map readings.
• Art and Design: Creating symmetric and angular shapes.
• Engineering: Mechanical joints and forces rely on angle calculations.

IX. Student Tips & Strategies for Success

 Time Management

• Day 1: Understand angle types and basic properties
• Day 2: Focus on transversal angles and linear pair
• Day 3: Practice diagrams, proofs, and PYQs

 Exam Preparation

• Practice drawing accurate diagrams
• Memorize all angle properties and relationships
• Solve previous year questions under timed conditions

 Stress Management

• Use colorful pens to mark corresponding angles while practicing
• Create a concept map linking all angle types

X. Career Guidance & Exploration

• For Classes 9–10:
  ➤ Foundation for geometry-heavy exams like NTSE, Olympiads, etc.
• For Classes 11–12 & Beyond:
  ➤ Relevant for careers in:
  • Architecture
  • Civil Engineering
  • Robotics
  • Industrial Design
  • Animation & Game Design
• Entrance Exams:
  ➤ JEE (Geometry section)
  ➤ NDA
  ➤ CUET – Math section
  ➤ Olympiads (SOF IMO, RMO)

XI. Important Notes

 Always label diagrams clearly in angle-based proofs.
 Understand rather than memorize properties—draw diagrams for clarity.
 Focus on angle pairs when transversals are involved.
 Review angle sum property regularly—it is foundational for higher geometry.