Vardaan
Class 9 Maths • Chapter 6 • NCERT + RS Aggarwal

Lines and Angles

Vardaan Learning Institute  |  School-Exam Focused Notes

📐 1. Basic Terms and Definitions

• Line segment: A part of a line with two endpoints. Denoted as AB.
• Ray: A part of a line with one endpoint. Denoted as $\overrightarrow{AB}$.
• Collinear points: 3 or more points lying on same line.
• Angle ($\angle$): Formed when two rays originate from the same endpoint (vertex).

Types of Angles

Acute Angle: $0° < x < 90°$
Right Angle: $x = 90°$
Obtuse Angle: $90° < x < 180°$
Straight Angle: $x = 180°$ (Straight line)
Reflex Angle: $180° < x < 360°$
(Reflex $\angle A = 360° - \angle A$)
Complete Angle: $x = 360°$

Pairs of Angles

Pair Name Definition Example/Condition
Complementary Two angles whose sum is 90° 30° and 60°
Supplementary Two angles whose sum is 180° 100° and 80°
Adjacent Angles Have common vertex, common arm, and non-common arms on different sides. They don't overlap.
Linear Pair Adjacent angles forming a straight line Sum = 180°
Vertically Opposite Formed when two lines intersect. They are always equal.

⚖️ 2. Axioms and Theorems (Intersecting Lines)

Axiom 6.1 (Linear Pair Axiom): If a ray stands on a line, then the sum of two adjacent angles so formed is 180°.
Axiom 6.2 (Converse): If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
📐 Theorem 6.1: Vertically Opposite Angles
Statement: If two lines intersect each other, then the vertically opposite angles are equal.

Given: Lines AB and CD intersect at O.
To prove: $\angle AOC = \angle BOD$ and $\angle AOD = \angle BOC$.
Proof:
1. Ray OA stands on line CD. $\therefore \angle AOC + \angle AOD = 180°$ (Linear Pair) ... (1)
2. Ray OD stands on line AB. $\therefore \angle AOD + \angle BOD = 180°$ (Linear Pair) ... (2)
3. Equating (1) & (2): $\angle AOC + \angle AOD = \angle AOD + \angle BOD$
4. Canceling $\angle AOD$ gives: $\angle AOC = \angle BOD$.
Similarly, $\angle AOD = \angle BOC$. Hence proved. (Very important for exams)
A B D C O Green = Vertically Opposite

🛤️ 3. Transversal and Parallel Lines

A line which intersects two or more lines at distinct points is called a transversal.
l m t 1 2 4 3 5 6 8 7

Angle Pair Naming (Crucial!)

Angle Type Pairs in figure above If l || m then...
Corresponding Angles (1,5); (2,6); (4,8); (3,7) Equal
Alternate Interior Angles (4,6); (3,5) Equal
Alternate Exterior Angles (1,7); (2,8) Equal
Consecutive Interior Angles (Co-interior) (4,5); (3,6) Supplementary (Sum=180°)
Corresponding Angles Axiom & Converses • If a transversal intersects two parallel lines, then each pair of corresponding angles is equal. (Axiom)
• Converse: If transversal intersects two lines making corresponding angles equal, the lines are parallel.
The same converse applies for alternate interior (equal) and co-interior (sum=180°).

📐 4. Lines Parallel to Same Line

Theorem 6.6: Lines which are parallel to the same line are parallel to each other.
If $l \parallel m$ and $n \parallel m$, then $l \parallel n$. (Uses corresponding angles axiom to prove).

🔺 5. Angle Sum Property of a Triangle

📐 Theorem 6.7: Triangle Angle Sum
Statement: The sum of the angles of a triangle is 180°.
Proof Strategy: Draw a line parallel to the base passing through the opposite vertex. Use alternate interior angles.
📐 Theorem 6.8: Exterior Angle Theorem
Statement: If a side of a triangle is produced, then the exterior angle so formed is equal to the sum of the two interior opposite angles.
$Ext\angle = \angle A + \angle B$
(Useful property: An exterior angle is always greater than either of its interior opposite angles).
✏️ Practice Questions — Lines and Angles (30 Questions)

Section A — 1 Mark / Calculations Easy

Q1. Find the complement of 35°.
Q2. Find the supplement of 105°.
Q3. Two adjacent angles on a straight line are in ratio 4:5. Find them.
Q4. Mention the types of angle: 89°, 91°, 180°, 200°.
Q5. Vertically opposite angles are 3x and x + 40. Find x.
Q6. What is the reflex angle of 60°?
Q7. If one angle of a linear pair is acute, the other must be?
Q8. In a triangle, if two angles are 40° and 60°, find the third.

Section B — 2/3 Mark Problems Medium

Q9. An angle is equal to its complement. What is it?
Q10. An angle is twice its supplement. Find it.
Q11. Lines AB and CD intersect at O. If $\angle AOC + \angle BOE = 70°$ and $\angle BOD = 40°$, find $\angle BOE$ and reflex $\angle COE$.
Q12. In $\triangle PQR$, if $\angle P - \angle Q = 42°$ and $\angle Q - \angle R = 21°$, find angles.
Q13. A transversal intersects two parallel lines. If ratio of interior angles on same side is 2:3, find angles.
Q14. Prove that bisectors of vertically opposite angles are in the same straight line.
Q15. If AB || CD, and a transversal intersects them finding alternate interior angles $3x-10$ and $2x+15$. Find x.
Q16. Find angles if exterior angle is 110° and interior opposite angles are in ratio 2:3.

Section C — Conceptual Proofs (NCERT) Medium

  1. Q17. Prove that if two lines intersect, vertically opposite angles are equal. (Write full proof).
  2. Q18. Prove that sum of angles of a triangle is $180°$.
  3. Q19. If a transversal intersects two parallel lines, prove that bisectors of alternate interior angles are parallel.
  4. Q20. If arms of one angle are parallel to arms of another angle, prove that they are either equal or supplementary.
  5. Q21. In figure, if AB || CD, EF $\perp$ CD and $\angle GED = 126°$, find $\angle AGE$, $\angle GEF$ and $\angle FGE$.
  6. Q22. In figure, OP, OQ, OR and OS are four rays. Prove that $\angle POQ + \angle QOR + \angle SOR + \angle POS = 360°$.

Section D — Hard / High Order Thinking Hard

  1. Q23. In $\triangle ABC$, the bisectors of $\angle B$ and $\angle C$ intersect at O. Prove that $\angle BOC = 90° + \frac{1}{2}\angle A$. (Standard Result)
  2. Q24. The sides AB and AC of $\triangle ABC$ are produced to points P and Q. Bisectors of $\angle PBC$ and $\angle QCB$ intersect at O. Prove $\angle BOC = 90° - \frac{1}{2}\angle A$.
  3. Q25. In figure, PQ || RS. $\angle MXQ = 135°$ and $\angle MYR = 40°$. Find $\angle XMY$. (Hint: Draw line through M parallel to PQ).
  4. Q26. If two parallel lines are intersected by a transversal, prove that the quadrilateral formed by bisectors of interior angles is a rectangle.
  5. Q27. Mirror Question: Two plane mirrors are placed parallel to each other. An incident ray strikes first mirror, reflects, strikes second mirror, and reflects back. Prove that the incident ray and final reflected ray are parallel. (Uses angle of incidence = angle of reflection).
  6. Q28. Prove: The sum of the exterior angles of any polygon (one at each vertex) is 360°. (Apply on triangle).
  7. Q29. PQ and RS are two mirrors placed parallel. If AB is incident, BC is reflected, CD is second reflection. Prove AB || CD.
  8. Q30. In a triangle, if $\angle A = 2\angle B$ and $\angle A - \angle C = 20°$, find all angles.
✅ Key Answers: Q1:55° | Q2:75° | Q3:80°,100° | Q5:x=20 | Q6:300° | Q9:45° | Q10:120° | Q12:95°,53°,32° | Q13:72°,108° | Q15:x=25 | Q16:44°,66° | Q21:126°,36°,54° | Q25:85°

📝 Quick Revision

  1. Linear Pair: Sum is 180°. Vertically Opposite: Always equal.
  2. Parallel Lines + Transversal:
    • Corresponding angles equal.
    • Alternate interior/exterior angles equal.
    • Co-interior angles sum to 180°.
  3. Triangles: Sum of interior angles = 180°.
  4. Exterior Angle Theorem: Exterior angle = sum of 2 opposite interior angles.
  5. Standard results: Incenter angle $\angle BOC = 90° + \frac{1}{2}\angle A$. Excenter angle $\angle BOC = 90° - \frac{1}{2}\angle A$.