This document defines and describes different types of lines, angles, and their relationships. It defines lines, line segments, rays, and different types of angles such as acute, obtuse, right, straight, and reflex. It also describes relationships between angles such as adjacent angles, complementary angles, supplementary angles, vertically opposite angles, corresponding angles, and alternate interior angles. Finally, it lists important axioms and theorems regarding lines and angles, such as axioms about rays on a line forming a 180 degree angle, and theorems about vertically opposite and alternate interior angles of parallel lines being equal.

1. Lines And Angles
Class-9
Done By : Smrithi Jaya

2. Types of lines
▪ Line
a line can be defined as a straight one-
dimensional figure that has no thickness and
extends endlessly in both directions
Line Segment
A line segment can be defined as a line with 2
end points
Ray
A ray can be defined as a line with one end
point
▪ Collinear points : points that lie on the same
line
Non collinear points : points that do not lie on
tha same line

3. Angles
▪ When two rays originate from the same end point
an angle is formed.
▪ The rays are called arms and the endpoint is called
vertex

4. Types of Angles
▪ AcuteAngle - 0° − 90°
▪ Obtuse Angle - 90° − 180°
▪ Right Angle - 90°
▪ Straight angle - 180°
▪ Complete angle - 360°
▪ Reflex angle - 180° − 360°
In the given figure :
𝒚 = 𝟑𝟎𝟎° 𝒊. 𝒆 𝒚 𝒊𝒔 𝒕𝒉𝒆 𝒓𝒆𝒇𝒍𝒆𝒙 𝒂𝒏𝒈𝒍𝒆
𝒙 = 𝟑𝟔𝟎 − 𝒚 = 𝟑𝟔𝟎 − 𝟑𝟎𝟎 = 𝟔𝟎°

5. Types of Angles

6. Adjacent angles
▪ Two angles are said to
be adjacent if they
have
▪ A common vertex
▪ A common arm
▪ Two non common
arms on different
sides of the common
arm
• Point B is the common
vertex
• BD is the common arm
• BA and BC are non-
common arms
• Therefore
< 1 𝑖𝑠 𝑎𝑑𝑗𝑎𝑐𝑒𝑛𝑡 𝑡𝑜 < 2

7. Complementary and
supplementary angles
▪ Complementary
angles
▪ Two angles whose
sum is 90° are called
complementary
angles
▪ SupplementaryAngles
▪ Two angles whose
sum is 180° are called
supplementary angles

8. Linear pairs and vertically
opposite angles
▪ Linear pair property or
staraight line property
▪ Vertically opposite
angles
If AC is a straight line,
then 𝑥 + 𝑦 = 180° or
< 𝐴𝐵𝐷 and < 𝐶𝐵𝐷 is a
linear pair
When AB and CD intersect
at O two pairs of vertically
opposite angles are formed
and are equal
i.e. < 1 = < 2
< 3 = < 4

9. Intersecting and
Non-intersecting lines
In this figure PQ and RS are
Intersecting lines
PQ and RS are parallel or
non - intersecting lines

10. Parallel lines and
transversal

11. Axioms and Theorems
AXIOMS THEOREMS
The axiom is a statement which is self
evident
theorem is a statement which is not
self evident
Axiom cannot be proven by any kind of
mathematical representation.
Theorem can be proved by
mathematical representation

12. AXIOMS
▪ Axiom 6.1 : if ray stands on a line the sum of two
angles formed is 180°
▪ 𝑥 + 𝑦 = 180°
▪ Axiom 6.2 : if sum of two angles is 180° then the
two non common arms form a line
▪ i.e AC is a line

13. AXIOMS
▪ Axiom 6.3 or corresponding angles axiom: if a
transversal intersects two parallel lines, then each
pair of corresponding angles is equal.
i.e. < 1 =< 5 , < 2 = < 6 , < 3 = < 7, < 4 =< 8
Axiom 6.4 : If a transversal intersects two
lines such that angles formed are corresponding then
the two lines are said to be parallel.

14. THEOREMS
▪ Theorem 1 : vertically opposite angles are
congruent
▪ Theorem 2: if a transversal intersects two parallel
lines , then each pair of alternate interior angles
are equal
▪ Theorem 3 : If a transversal intersects two parallel
lines such that a pair of alternate interior angles is
equal then the two lines are parallel.
▪ Theorem 4 : If a transversal intersects two parallel
lines then each pair of co-interior angles are
supplementary.

15. THEOREMS
▪ Theorem 5 : If a transversal intersects two parallel
lines such that a pair of co interior angles are
supplementary then the two lines are parallel.
▪ Theorem 6 : Lines which are parallel to the same
line are parallel to each other.
▪ Theorem 7:The sum of the angles of a triangle is
180°.
▪ Theorem 8 or Exterior angle theorem : If a side of a
triangle is produced then the exterior angle so
formed is equal to sum of interior opposite angles.

16. Done By:
Smrithi Jaya