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Class 9 Maths notes for Lines And Angles





We have already studied point,line,line segment in previous chapter.
Just a recap to help you
a) Point: It is denoted by a single dot on the paper and it is represented by English alphabet. It has no length ,breadth and thickness
b) Line: A line is straight and has no curves. It extends in both the direction and has no end point. We have infinite lines through a point while we will have only one line through two points
c) Line Segment:If we take a portion of a line ,then it is called line segment and it has length and two end points. It can be measured
d) Ray: If a line has one end point and it can extent in other direction,then it is called a ray
e) Congruent segments: If two line segments are equal then they are called congruent
Now we will extend our knowledge to angles in this chapter

What is angle

An angle is formed by two rays with a common endpoint. The Common end point is known as the vertex of the angle and the rays as the sides, sometimes as the legs and sometimes the arms of the angle
angle defination We denote the angle by symbol $\angle $
If A is the Vertex,then angle could be represented as $\angle A$

Types of Angle

As the angle size is increased,it is termed with different name.
acute angle
Obtuse angle
Right angle
Reflex angle
Straight angle

Degree and Radian

They both are unit of measurement of angles
Radian: A unit of measure for angles. One radian is the angle made at the centre of a circle by an arc whose length is equal to the radius of the circle.
Degree: If a rotation from the initial side to terminal side is (1/360) of a revolution, the angle is said to have a measure of one degree, written as 1°. A degree is divided into 60 minutes, and a minute is divided into 60 seconds . One sixtieth of a degree is called a minute, written as 1", and one sixtieth of a minute is called a second, written as 1'.
Thus, 1° = 60', 1' = 60"
Relation between Degree and Radian
2π radian = 360 ° π radian= 180 ° 1 radian= (180/π) °
Degree30°45°60°90°120°180°360°
Radianπ/6π/4π/3π/22π/3π


Congruence Angle

Two angles are said to be congruent if they are having same measure
$\angle A=30^0$
$\angle B=30^0$
then
$\angle A=\angle B$
They are congruent

Adjacent Angles

  • Two angles are called adjacent angles if they share the same vertex,they have a common arm. The second arm of the one angle is one side and second arm of other angle is on the other side
  • Example given below
    Adjacent angle

Complimentary Angles

  • Two angles , the sum of whose measures is $90^0$ is called Complimentary Angles.Each of these Complimentary Angles are called the complement of each other
  • Example given below
    Complimentary Angles
    Here the sum of angles =$90^0$

Supplementary Angles

  • Two angles , the sum of whose measures is $180^0$ is called Supplementary Angles.Each of these Supplementary Angles are called the supplement of each other
  • Example given below
    Supplementary Angles
    Here the sum of angles =$180^0$

Linear Pair Axioms

  • If a ray stands on a line, then the sum of the adjacent angles so formed is $180^0$
  • And If the sum of the adjacent angles is $180^0$,then the non common arms of the angles form a line
  • Linear Pair
Theorem Based on Linear Pair Axiom
The sum of all the angles around a point is $360^0$
the sum of all the angles around a point
$\angle 1 + \angle 2 + \angle 3 + \angle 4 =360^0$


Vertically Opposite angles

If two lines intersect with each other, then vertically opposite angles are equal
Vertically Opposite angles

Transversal across the parallel Lines

If the transversal intersect two parallel lines as shown in below figure
Transversal across the parallel Lines
Important Take away from the figure
  1. We can see following angles as depicted in the figure above
    $\angle 1,\angle 2,\angle 3,\angle 4$ on the first parallel line
    and $ \angle 5,\angle 6,\angle 7,\angle 8$ on the second parallel line
  2. The angles 1,2,6,7 are called exterior angles while the angles 4,3,5,8 are called interior angles
    exterior angles = $\angle 1, \angle 2, \angle 6 , \angle 7$
    interior angles = $\angle 4, \angle 3, \angle 5 , \angle 8$
  3. Corresponding Angles The angles on the same side of the Transversal are known as Corresponding angles
    And Corresponding Angles axiom states that
    Corresponding Angles
  4. Each pair of alternate interior angles are equal
    Each pair of alternate interior angles are equal
  5. Each pair of interior angles on the same side of the transversal is supplementary
    Each pair of interior angles on the same side of the transversal  is supplimentary

Converse of Transversal across the parallel Lines

If a transversal intersect two lines such that either
  1. Any one pair of corresponding angles are equal
  2. any one pair of alternate interior angles are equal
  3. any one pair of interior angles on the same side of the transversal is supplementary
Then the two lines are parallel

Parallel lines Note

Lines which are parallel to a given line are parallel with each other

Angle sum property of Triangles

The sum of the angles of the triangle
  • The sum of the angles of the triangle is $180^0$
  • if the side of the triangle is produced ,the exterior angle formed is equal to the sum of the opposite interior angle
    the exterior angle formed is equal to the sum of the opposite interior angle

Solved examples

Question 1) The angles of a triangle are in the ratio 5 : 3 : 7. The triangle is :

(a) an acute angled triangle
(b) an obtuse angled triangle
(c) a right triangle
(d) an isosceles triangle

Solution (a) an acute angled triangle
Let the angles be 5x, 3x and 7x. Using angle sum property, $5x + 3x + 7x = 180^0$
$15x = 180^0$
$x = 12^0$
Hence angles $60^0, 36^0, 84^0$ .SO it is an acute angle triangle.

Question 2) An exterior angle of a triangle is $130^0$ and its two interior opposite angles are equal. Each of the interior angle is equal to:
(a) $45^0$
(b) $65^0$
(c) $75^0$
(d) $35^0$
Solution
Let x be the two angles equal
then $2x + 50=180$
$x=65$

Question 3) True and false statement
(a) A triangle can have two right angles
(b) A triangle can have two obtuse angles
(c) A triangle can have two acute angles
(d) A triangle can have all angles less than $60^0$
(e) A triangle can have all angles more than $60^0$
(f) A triangle can have all angles equal to $60^0$
g) The two acute angles in every right triangle are complementary.

Solution
(a) False. Since the sum of three angles of a triangle is $180^0$. Sum of two right angles is 180, such triangle is not possible.
(b) True Because the sum of two obtuse angles will become greater than $180^0$. Such a triangle is not feasible.
(c) True a triangle can have two acute angles. Since the sum of two acute angle is less than $180^0$, the third angle will have the remaining value.
(d) False Since sum of all angles less than $60^0$ is still less than $180^0$. Such a triangle is not possible.
(e) False Since sum of angles (each angle is greater than $60^0$) exceed $180^0$, such a triangle is not possible.
(f) True Sum of angles (each = $60^0$) is exactly equal to $180^0$, such a triangle is possible. It makes an equilateral triangle.
g) True
Crossword for Line and Angle
Crossword for Line and Angle
Across
5. The angles on the same side of the Transversal are known as
8. Unit of angle
9. If two lines intersect with each other, then _______ opposite angles are equal
Down
1. Angles whose sum is $180^0$
2. Angle( $90^0 < \theta < 180^0$ )
3. Angle( $0^0 < \theta < 90^0$ )
4. Angles which share one side
6. Angle ($\theta=90^0$
7. Angle ($180^0 < \theta < 360^0$ )
10. Angles whose sum is $90^0$

Quiz Time

Question 1Find the value of x for which line l and m are parallel ?

A. $136^0$
B. $44^0$
C. $46^0$
D. none of the above
Question 2Find the value of z in the above questions
A. $136^0$
B. $44^0$
C. $46^0$
D. none of the above
Question 3 Find the value of y
A. $136^0$
B. $44^0$
C. $46^0$
D. none of the above
Question 4 The angles of a triangle are in the ratio 2 : 3 : 4. Find the angles of the triangle
A. $80^0,120^0,160^0$
B. $20^0,30^0,40^0$
C. $40^0,60^0,80^0$
D. None of the above
Question 5 The lines AB, CD and EF are three lines concurrent at O as shown below in figure

Find the value of $\angle EOC$> A. $12^0$
B. $30^0$
C. $80^0$
D. $40^0$
Question 6 In the above question, find the angle $ \angle COB$ ?
A. $12^0$
B. $30^0$
C. $20^0$
D. $40^0$


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Reference Books for class 9 Math

Given below are the links of some of the reference books for class 9 Math.

  1. Mathematics for Class 9 by R D Sharma One of the best book for studying class 9 level mathematics. It has lot of problems to be solved.
  2. Secondary School Mathematics for Class 9 by R S Aggarwal This is also as good as R.D. Sharma. Either this or the book by R.D. Sharma will do. I find book R.S. Aggarwal little bit more challenging than the one by R.D. Sharma.
  3. Pearson IIT Foundation Series - Maths - Class 9 Buy this book if you want to challenge yourself further and want to prepare for JEE foundation.
  4. Pearson IIT Foundation Physics, Chemistry & Maths combo for Class 9 Only buy if you are prepared to study extra topics and want to take your studies a step further. You might need help to understand topics in these books.

You can use above books for extra knowledge and practicing different questions.



Class 9 Maths Class 9 Science





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