| --- |
| source: ncert |
| topic: geometry |
| class_level: class_9 |
| chapter: ch6 |
| difficulty: intermediate |
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| |
| 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. |
