source: ncert
topic: geometry
class_level: class_9
chapter: ch6
difficulty: intermediate
Class 9 | Ch6: Lines and Angles KEY THEOREMS (with proofs): Theorem 1: If two lines intersect, vertically opposite angles are equal. Proof: angle 1 + angle 2 = 180° (linear pair). angle 3 + angle 2 = 180° (linear pair). So angle 1 = angle 3. Similarly angle 2 = angle 4. Theorem 2: Sum of angles on a straight line = 180°. Theorem 3: If transversal cuts parallel lines: (a) Corresponding angles are equal. (b) Alternate interior angles are equal. (c) Co-interior angles are supplementary (sum=180°). Theorem 4 (Converse): If corresponding angles are equal → lines are parallel. ANGLE SUM PROPERTY OF TRIANGLE: Sum of angles of triangle = 180°. Proof using parallel line through vertex. Draw BC. Draw line PQ through A parallel to BC. Angle PAB = Angle ABC (alternate interior). Angle QAC = Angle ACB (alternate interior). Angle PAB + Angle BAC + Angle QAC = 180° (straight line PQ). Therefore Angle ABC + Angle BAC + Angle ACB = 180°. EXTERIOR ANGLE THEOREM: Exterior angle of triangle = sum of two non-adjacent interior angles. SOLVED EXAMPLES: Example 1: Two parallel lines cut by transversal. One angle=65°. All 8 angles: 65°, 115°, 65°, 115°, 65°, 115°, 65°, 115°. Alternate interior=65° (equal). Co-interior=115° (supplementary). Corresponding=65° (equal). Example 2: Prove vertically opposite angles are equal. Lines AB and CD intersect at O. Angle AOC + Angle AOD = 180° (linear pair on line CD). Angle BOD + Angle AOD = 180° (linear pair on line AB). Therefore Angle AOC = Angle BOD. (both equal 180° - Angle AOD) COMMON MISTAKES: Co-interior angles are SUPPLEMENTARY (sum=180°), not equal. Exterior angle = sum of REMOTE interior angles, NOT the adjacent interior angle.