Lines, Angles, Polygons and Symmetry
Lines, Angles, Polygons and Symmetry
Geometry starts with angles: pairs that add to 90° or 180°, the angles a transversal makes with two parallel lines, and the angle sums of polygons. NTPC sets algebraic angles on parallel lines, interior and exterior angles of regular polygons with their diagonals, and lines of symmetry of figures and letters.
1. Angle Facts
| Pair or rule | Relation |
|---|---|
| Complementary angles | Sum 90° |
| Supplementary angles (linear pair) | Sum 180° |
| Vertically opposite angles | Equal |
| Angles round a point | Sum 360° |
| Parallel lines: corresponding angles | Equal |
| Parallel lines: alternate interior angles | Equal |
| Parallel lines: co-interior (same-side interior) angles | Sum 180° |
| Bisectors of two co-interior angles | Meet at 90° |
2. Polygon Facts (n sides)
Sum of interior angles = (n − 2) × 180° Sum of exterior angles = 360°
Regular polygon: each exterior angle = 360°/n, each interior angle = 180° − 360°/n
Number of diagonals = n(n − 3)/2
| Regular polygon | n | Exterior | Interior | Diagonals |
|---|---|---|---|---|
| Pentagon | 5 | 72° | 108° | 5 |
| Hexagon | 6 | 60° | 120° | 9 |
| Octagon | 8 | 45° | 135° | 20 |
| Nonagon | 9 | 40° | 140° | 27 |
| Decagon | 10 | 36° | 144° | 35 |
| Dodecagon | 12 | 30° | 150° | 54 |
3. Symmetry Table
| Figure | Lines of symmetry | Order of rotational symmetry |
|---|---|---|
| Scalene triangle | 0 | 1 |
| Isosceles triangle | 1 | 1 |
| Equilateral triangle | 3 | 3 |
| Rectangle, rhombus | 2 | 2 |
| Square | 4 | 4 |
| Parallelogram (general) | 0 | 2 |
| Isosceles trapezium | 1 | 1 |
| Regular n-gon | n | n |
| Circle | Infinitely many | Infinite |
✗ The exterior angle of a regular polygon is 30°, so it has 30 sides | ✓ n = 360/30 = 12
✗ A parallelogram has 2 lines of symmetry | ✓ It has none; it only has rotational symmetry of order 2
हिंदी नोट: समांतर रेखाओं को तिर्यक रेखा काटे तो एकांतर कोण बराबर और एक ही ओर के अंतः कोणों का योग 180° होता है। सम बहुभुज का बाह्य कोण 360°/n और विकर्णों की संख्या n(n − 3)/2 होती है।
Exam Pointer: Verified NTPC patterns: algebraic angles on two parallel lines cut by a transversal (March 2026 Graduate CBT-1) and a regular polygon's sides and diagonals from its interior angle (August 2025 UG CBT-1). Complement-supplement and symmetry questions had no verified NTPC shift and are tagged syllabus-based.
Pariksha Pattern: Every Way NTPC Asks This Topic
Pattern 1: Parallel lines and a transversal with algebraic angles
[PYQ: NTPC Graduate CBT-1 16-Mar-2026 Shift-2]
EXAM LEVEL
Q. Two parallel lines are cut by a transversal. Two co-interior angles are (3x + 20)° and (2x + 10)°. Find both angles.
Co-interior angles add to 180°: 5x + 30 = 180, so x = 30. The angles are 110° and 70°.
Answer: 110° and 70°
EXAMATLAS LEVEL
Q. AB ∥ CD and a transversal meets AB at M and CD at N, with B and D on the same side. ∠BMN = (3x + 10)° and ∠MND = (2x + 20)°. Find ∠AMN, and the angle at which the bisectors of ∠BMN and ∠MND meet.
∠BMN and ∠MND are co-interior, so 5x + 30 = 180 and x = 30: ∠BMN = 100° and ∠MND = 80°. ∠AMN = 180° − 100° = 80° (it also equals the alternate angle ∠MND). The half-angles 50° and 40° add to 90°, so the bisectors meet at 90°.
Answer: 80°; 90°
Pattern 2: Complements and supplements
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. An angle is 24° more than its complement. Find the angle.
x = (90 − x) + 24 gives 2x = 114, so x = 57°.
Answer: 57°
EXAMATLAS LEVEL
Q. The supplement of an angle is four times its complement. Find the angle, and the angle whose supplement is 50° more than three times its complement.
180 − x = 4(90 − x) gives 3x = 180, so x = 60°. For the second, 180 − y = 3(90 − y) + 50 gives 2y = 140, so y = 70°.
Answer: 60°; 70°
Pattern 3: Regular polygons: sides, angles and diagonals
[PYQ: NTPC UG CBT-1 22-Aug-2025 Shift-2]
EXAM LEVEL
Q. Each interior angle of a regular polygon is 150°. Find the number of sides and diagonals.
Exterior angle = 30°, so n = 360/30 = 12. Diagonals = 12 × 9/2 = 54.
Answer: 12 sides; 54 diagonals
EXAMATLAS LEVEL
Q. The interior angle of a regular polygon is 4 times its exterior angle. Find the number of sides, the sum of its interior angles and its number of diagonals.
Exterior + interior = 180° and the ratio is 1 : 4, so the exterior angle = 36° and n = 10. Sum of interior angles = 8 × 180° = 1,440°. Diagonals = 10 × 7/2 = 35.
Answer: 10; 1,440°; 35
Pattern 4: Lines of symmetry and rotational symmetry
[Pattern: syllabus-based, PYQ-style]
EXAM LEVEL
Q. How many lines of symmetry does a regular octagon have, and what is its order of rotational symmetry?
A regular n-gon has n lines and order n: 8 and 8.
Answer: 8 and 8
EXAMATLAS LEVEL
Q. Among the capital letters A, B, H, N, S, X and Z (in plain block form), which have both a line of symmetry and rotational symmetry of order 2?
A has only a vertical line and B only a horizontal line. N, S and Z look the same after a half turn but have no line of symmetry. H and X have two lines each and also look the same after a half turn.
Answer: H and X
60-Second Revision
- Parallel lines: alternate and corresponding angles equal; co-interior add to 180°.
- Regular polygon: exterior 360°/n; interior + exterior = 180°.
- Diagonals = n(n − 3)/2; interior sum = (n − 2) × 180°.
- Regular n-gon: n lines of symmetry, rotational order n.