Lines & Angles — Class 9 (Interactive)

Complete NCERT coverage — interactive diagrams, all key theorems, 5 solved examples & 5 practice problems.

Essentials — Definitions & Types

Line: Straight path extending infinitely in both directions. Ray: Starts at a point and extends infinitely in one direction. Line segment: Part of a line with two endpoints.

Angle: Formed by two rays with a common endpoint (vertex). Measured in degrees (°).

Types of angles

  • Acute < 90°
  • Right = 90°
  • Obtuse > 90° and < 180°
  • Straight = 180°
  • Reflex > 180°

Angle pairs

  • Adjacent angles — share a side and vertex
  • Linear pair — adjacent and sum to 180°
  • Vertically opposite — equal
  • Corresponding, alternate interior/exterior (when parallel lines are cut by transversal)

Interactive Diagrams — Hover to highlight

Move your mouse (or tap) over labelled lines/angles to highlight the related parts and read the short explanation.

∠1 ∠2 ∠3 ∠4 Vertically opposite angles: ∠1 = ∠2, ∠3 = ∠4 ∠A (alt int) ∠B (alt int) If p1 ∥ p2, alternate interior angles are equal: ∠A = ∠B
Highlight on hover

All Important Theorems (with short proofs/sketches)

  1. Vertically opposite angles are equal. (Sketch) Two straight lines intersect → linear pairs equal → hence vertical angles equal.
  2. Linear pair of angles are supplementary. (Sum = 180°)
  3. If a transversal intersects parallel lines, then corresponding angles are equal. (Use alternate interior equality and congruent triangles idea.)
  4. Alternate interior angles are equal when lines are parallel.
  5. Co-interior angles are supplementary.

These are NCERT core theorems — the interactive diagrams above let you visualise each claim.

5 Solved Examples (step-by-step)

1. Find the value of x if two lines intersect and produce vertically opposite angles (3x + 10)° and (5x − 6)°.
Set equal: 3x + 10 = 5x − 6 ⇒ 16 = 2x ⇒ x = 8.
2. In diagram, two parallel lines cut by transversal produce consecutive interior angles (4x + 20)° and (3x − 5)°. Find x.
Consecutive interior sum to 180 ⇒ (4x + 20) + (3x − 5) = 180 ⇒ 7x + 15 = 180 ⇒ 7x = 165 ⇒ x = 23.571... (or 165/7).
3. If corresponding angles are (2x + 5)° and (5x − 10)°, lines are parallel — find x.
Set equal: 2x + 5 = 5x − 10 ⇒ 15 = 3x ⇒ x = 5.
4. If angle of a linear pair is (6x − 14)° and other is (2x + 10)°, find x.
Sum = 180 ⇒ (6x − 14) + (2x + 10) = 180 ⇒ 8x − 4 = 180 ⇒ 8x = 184 ⇒ x = 23.
5. Two lines intersect. One acute angle is (7x − 20)° and its adjacent angle is (3x + 10)°. Find x.
Adjacent angles sum to 180? Not necessarily; they might be linear pair if they are on same straight line. If they are adjacent and form straight line, set sum = 180 ⇒ (7x − 20) + (3x + 10) = 180 ⇒ 10x −10 = 180 ⇒ 10x = 190 ⇒ x = 19.

5 Practice Questions

1. Two lines intersect. Vertically opposite angles are (4x + 8)° and (6x − 24)°. Find x and the angles.
2. If alternate interior angles are (3x + 15)° and (5x − 1)°, find x and decide if lines are parallel.
3. A linear pair angles are in ratio 3:2. Find the angles.
4. Two parallel lines are cut by a transversal forming corresponding angles (2x + 12)° and (4x − 8)°. Find x.
5. Find x if three adjacent angles around a point on a straight line are (x + 10)°, (2x − 5)°, and (3x − 25)°.

Answers (brief)

  1. 4x + 8 = 6x − 24 ⇒ 32 = 2x ⇒ x = 16. Angles = 4(16)+8 = 72° and 108° (vertical pairs accordingly).
  2. 3x + 15 = 5x − 1 ⇒ 16 = 2x ⇒ x = 8 ⇒ angles 39° and 39° → lines are parallel.
  3. Angles are 3k and 2k, sum 180 ⇒ 5k = 180 ⇒ k = 36 ⇒ angles 108° and 72°.
  4. 2x + 12 = 4x − 8 ⇒ 20 = 2x ⇒ x = 10.
  5. Sum = 180 ⇒ (x+10)+(2x−5)+(3x−25)=180 ⇒ 6x −20 =180 ⇒ 6x=200 ⇒ x=200/6=100/3 ≈33.33.