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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. |