Students can go through AP 9th Class Maths Notes Chapter 6 Lines and Angles to understand and remember the concepts easily.
Class 9 Maths Chapter 6 Notes Lines and Angles
→ Line segment : A part or portion of a line with two end points (terminal points) is a called a line segment.
Ex : 
Here ‘l’ is a line, \(\overline{\mathrm{AB}}\) is a line segment and is denoted by \(\overline{\mathrm{AB}}\) and A, B are two end points of \(\overline{\mathrm{AB}}\). \(\overline{\mathrm{AB}}\) = \(\overline{\mathrm{BA}}\), AB is length of \(\overline{\mathrm{AB}}\) (or) \(\overline{\mathrm{BA}}\).
→ Ray : A part (one side portion) of a line is called a ‘Ray’. A ray will have only one end point.
Ex : 
Here ‘l’ is a line and B, O, A are different points, then a part starting from ‘O’ through “A” is called Ray OA and is denoted by \(\overrightarrow{\mathrm{OA}}\). Here ‘O’ is end point and the part from ‘O’ through ‘B’ is called a ray OB and is denoted by \(\overrightarrow{\mathrm{BO}}\) (or) \(\overrightarrow{\mathrm{OB}}\). Here ‘O’ is end point.
Note : \(\overrightarrow{\mathrm{OB}}\), \(\overrightarrow{\mathrm{BO}}\) are not same. Observe the following figure, and
So \(\overline{\mathrm{OB}}\) and \(\overline{\mathrm{BO}}\) are not same.
→ Collinear points: The points which are on a same line are called collinear points.
Ex : 
Here A, B, C, D, E, F, ……… are on same line. Hence they are collinear.
→ Non-collinear : If three or more points lie on the same line, they are called collinear points, otherwise they are called non-collinear points.
Ex : 
Here A, B, D are on same line, so they are called collinear points.
Here A, B, C are not on same line. Hence non-collinear.
Also B, C, D are non-coliinear points. A, B, C, D, E are non-collinears.
→ Angle: We know that, an angle is formed when two rays have common end point.
Ex : \(\overrightarrow{\mathrm{OA}}\), \(\overrightarrow{\mathrm{OB}}\) here two rays \(\overrightarrow{\mathrm{OA}}\), \(\overrightarrow{\mathrm{OB}}\) have common end point ‘O’. So an angle is formed at ‘O’ as shown below.
This common end point of two rays is called “vertex” of the angle and two rays having the angle are called arms of angle. Here in the above figure,
i) Rays \(\overrightarrow{\mathrm{OA}}\), \(\overrightarrow{\mathrm{OB}}\) have a common end point.
ii) So, for ∠AOB (or) ∠BOA forms
iii) ‘O’ is vertex of ∠AOB (or) ∠BOA.
iv) \(\overrightarrow{\mathrm{OA}}\), \(\overrightarrow{\mathrm{OB}}\) are two rays of ∠AOB (or) ∠BOA.
→ Acute angle : If the angle at a point (or between two arms) is in between ‘O’ and 90°, then the angle is called acute angle. x° is an acute angle if 0° < x < 90°.
→ Right angle : If the angle in between two arms is 90°, then the angle is called right angle.
here ∠P = 90°
→ Obtuse angle : If the angle in between two arms is more than 90° but less than 180°, then the angle is called obtuse angle.
If 90° < θ < 180°, then ’θ’ is called obtuse angle.
→ Straight angle : An angle between two opposite rays is called straight angle. It measures 180°.
So an angle equal to 180° is called straight angle.
∠BOA (or) ∠AOB = 180° hence straight angle.
→ Reflex angle : An angle more than 180° but less than 360° is called Reflex angle.
Here 180° < Q < 360°
So, ∠AQB (or) ∠BQA is called reflex angle.
→ Complementary angles : Two angles P, Q are said to be complementary angles, if their sum is equal to 90°.
i.e. ∠P + ∠Q = 90°, then P, Q are complementary angles.
Ex : (30°, 60°) (20°, 70°) (1°, 89°) are pairs of complementary angles.
→ Supplementary angles: Two angles ∠A, ∠B are said to be supplementary if their sum is equal to 180°.
- If ∠A + ∠B = 180°, then ∠A, -∠B are called supplementary angles.
- (60°, 120°) (30°, 150°) (170°, 10°) etc., are called pairs of supplementary angles.
- If two angles together make a straight angle, then they are supplementary angles.
∠AOB = ∠P, ∠BOC = ∠Q and
∠AOB + ∠BOC = ∠P + ∠Q = 180°
Hence ∠P, ∠Q are supplementary angles.
→ Adjacent angles : Two angles are said to be adjacent angles. If they have
i) a common vertex
ii) a common arm
iii) non-common arms are on either side of their common arm.
Ex:
i) 
∠POQ, ∠QOR are adjacent angles, because they have common vertex ‘O’ and common arm \(\overrightarrow{\mathrm{OQ}}\) and noncommon arms \(\overrightarrow{\mathrm{OP}}\), \(\overrightarrow{\mathrm{OR}}\) are on different sides of common arm \(\overrightarrow{\mathrm{OQ}}\).
ii) 
Here ∠APB, ∠BPC are adjacent angles.
iii) But
∠KOL, ∠KOJ are not adjacent angles, why because their non¬common arms \(\overrightarrow{\mathrm{OL}}\), \(\overrightarrow{\mathrm{OJ}}\) are pot on either side of their common arm \(\overrightarrow{\mathrm{OR}}\).
Here both \(\overrightarrow{\mathrm{OJ}}\), \(\overrightarrow{\mathrm{OL}}\) are on same side of their common arm \(\overrightarrow{\mathrm{OK}}\).
Hence they are not adjacent angles.
→ Linear pair angles : If two adjacent angles are supplementary, then they are called linear pair angles.
Ex:
∠AOB, ∠BOC are adjacent and they are supplementary (∵ they form straight angle together). Hence linear pair angle.
→ Vertically opposite angles : When two lines intersect at a point (O) then the pairs of non-adjacent angles at the intersection point are called pairs of vertically opposite angles.
Then ∠AOB, ∠COD are non adjacent angles formed by two lines \(\overrightarrow{\mathrm{AC}}\), \(\overrightarrow{\mathrm{BD}}\).
Hence ∠AOB, ∠COD, ∠AOD, ∠BOC are vertically opposite angles.
→ Intersecting lines : If two lines meet at a point, then those lines are called intersecting lines.
Here l, m are meeting each other at a point “O”. So they (l, m) are called intersecting lines and the point at which they meet each other is called “intersecting point”.
(Or)
If two lines have a common point, then they are called intersecting lines. And the common point is called point, of intersection.
→ Non-intersecting lines (Parallel lines) : If two lines on a plane don’t intersect each other then they are called nonintersecting lines (or) parallel lines. Here common perpendiculars (distance) at different points on these parallel lines are same.
(Same distances). Hence parallel lines. We use symbol (//) for parallel. So l // m.
→ Relations – pairs of angles : If a ray stands on a line, then the sum of adjacent angles so formed is 180° (or) supplementary.
Ex:
\(\overleftrightarrow{\mathrm{AB}}\) is the line.
\(\overleftrightarrow{\mathrm{OS}}\) is the ray stand on line.
∠s, ∠t are a pair of adjacent angles,
so formed.
then, ∠t + ∠s = 180°
Hence supplementary (or) linear pair of angles.
→ Vice-versa (Converse) :
i) If the sum of the adjacent angles is 180°, then the ray (which gives adjacent angles) stands on a line (that is non-common arms form a line).
ii) If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.
Given ∠p + ∠q = 180°, formed by, \(\overrightarrow{\mathrm{OC}}\) then non-common arms of ∠AOC, ∠BOC are \(\overrightarrow{\mathrm{OA}}\) and \(\overrightarrow{\mathrm{OB}}\) which forms a line \(\overleftrightarrow{\mathrm{AOB}}\).
iii) If two lines intersect each other, then the vertically opposite angles are equal.
Proof :
Let \(\overleftrightarrow{\mathrm{AOB}}\) is a line
and \(\overleftrightarrow{\mathrm{COD}}\) is another line
and \(\overleftrightarrow{\mathrm{AOB}}\), \(\overleftrightarrow{\mathrm{COD}}\) intersects each other at the point “O”.
Then let the angles formed at ‘O’ are ∠1, ∠2, ∠3, ∠4 as shown in figure, then∠1 + ∠2 = 180° [∵ adjacent angles formed by ray on line]
and ∠2 + ∠3 = 180° [∵ adjacent angles formed by ray on line]
∴ ∠1 + ∠2 = ∠2 + ∠3
⇒ ∠1 = ∠3
We know ∠1, ∠3 are called vertically opposite angles.
Hence ∠1 = ∠3, so vertically opposite angles are equal and also from the above figure,
∠1 + ∠4 = 180° (adjacent angles formed by a ray on a line)
and ∠1 + ∠2 = 180° (adjacent angles formed by a ray on a line)
∴ ∠1 + ∠4 = ∠1 + ∠2 ⇒ ∠4 = ∠2
∠4, ∠2 are vertically opposite angles. Hence vertically opposite angles formed by intersecting two lines are always equal.
→ Different angles and their relation :
Angles formed by a transversal on parallel lines.
• Lines which are parallel to the same line are parallel to each other.
Explanation : Let l, m are parallel to a line ‘n’, then l//m.
Angles – formed by transversesal.
i) Corresponding angles: The pairs (∠1, ∠5), (∠2, ∠6), (∠3, ∠7) and (∠4, ∠8) are called corresponding angles.
Corresponding angles axiom :
Note: The corresponding angles formed on parallel lines by a transversal are equal and the converse also true.
It means,
if the corresponding angles formed on two lines by a transversal are equal, then those lines are parallel.
In the above figure, .
If l // m, then ∠1 = ∠5; ∠2 = ∠6; ∠3 = ∠7; ∠4 = ∠8 and if ∠1 = ∠5 (or) ∠2 = ∠6 or ∠3 = ∠7 or ∠4 = ∠8, then (l // m).
→ Interior angles: In the above figure, ∠3, ∠4, ∠5, ∠6 are called interior angles.
→ Alternate interior angles : A pair of in-terior angles that are not adjacent and on either side of transversal are called alternate interior angles.
∠3, ∠5 are alternate interior angles and ∠4, ∠6 are alternate interior angles.
If l // m, then ∠3 = ∠5 and ∠4 = ∠6 and if ∠3 = ∠5 and ∠4 = ∠6, then l // m.
→ Exterior angles : ∠1, ∠2, ∠7, ∠8 are exterior angles.
→ Alternate exterior angles : A pair of exterior angles that are not adjacent and on either side of transversal are called alternate exterior angles.
∠1, ∠7 are alternate exteriors and also ∠2, ∠8 ar also alternate exteriors.
If ∠1 = ∠7 (or) ∠2 = ∠8, then l // m.
If l // m, then ∠1 = ∠7 and ∠2 = ∠8.
→ A ray is a part of line. It begins at a point and goes on endlessly in a specified direction.
→ A part of a line with two end points is called a line segment.
→ Points on the same line are called collinear points.
→ The angle is formed by rotating a ray from an initial position to a terminal position.
→ One complete rotation makes an angle 360°.
→ Angles are named according to their measure.
Obtuse angle 90° < x < 180°.
→ Straight angle y = 180°
→ Reflex angle 180° < z < 360°
→ If two lines have no common points, they are called parallel lines. In the figure l // m.
→ If two lines have a common point then they are called intersecting lines.
→ Three or more lines meet at a point are called concurrent lines.
→ Two angles are said to be supplementary angles if their sum is 180°.
E.g.: (100°, 80°), (110°, 70°), (120°, 60°), (179°, 1°), (90°, 90°) etc.
→ The supplementary angle to x° is given by (180° – x°).
→ Two angles are said to be complementary if their sum is 90°.
→ The complementary angle to x° is (90° – x°).
E.g.: (89°, 1°), (70°, 20°), (60°, 30°) etc.
→ Two angles are said to be form a pair of adjacent angles if they have a common arm and lie on the either sides of the common arm.
∠1 and ∠2 are a pair of adjacent angles with OB as their common arm.
→ A pair of adjacent angles are said to be a linear pair of angles if their sum is 180°. ∠1 + ∠2 = 180°
→ When a pair of lines meet at a point, they form four angles. The two pairs of angles which have no common arm are called vertically opposite angles.
In the figure (∠1, ∠3) and (∠2, ∠4) are the pairs of vertically opposite angles.
→ When two lines intersect, the pairs of vertically opposite angles thus formed are equal a = c and b = d. (from the figure)
→ When a pair of lines intersected by a transversal, there forms eight angles.
→ When a pair of parallel lines intersected by a transversal the pairs of a) alternate interior angles b) corresponding angles c) alternate exterior angles are equal and the interior / exterior angles on the same side of the transversal are supplementary.
→ Lines which are parallel to same line are parallel to each other.
→ The sum of the interior angles of a triangle is 180°.
→ If one side of a triangle is produced, then the exterior angle thus formed is equal to the sum of the two interior opposite angles.