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

More can be read on the below link

Euclid Geometry

Now we will extend our knowledge to angles in this chapterJust a recap to help you

a.

b.

c.

d.

e.

More can be read on the below link

Euclid Geometry

- What is angle
- Congruence Angle
- Adjacent Angles
- Complimentary Angles
- Supplementary Angles
- Linear Pair Axioms
- Transversal across the parallel Lines
- Angle sum property of Triangles

We denote the angle by symbol $\angle $

If A is the Vertex,then angle could be represented as $\angle A$

Thus, 1° = 60', 1' = 60"

2π radian = 360 ° π radian= 180 ° 1 radian= (180/π) °

Degree | 30° | 45° | 60° | 90° | 120° | 180° | 360° |

Radian | π/6 | π/4 | π/3 | π/2 | 2π/3 | π | 2π |

$\angle A=30^0$

$\angle B=30^0$

then

$\angle A=\angle B$

They are congruent

- 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

- 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

Here the sum of angles =$90^0$

- 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

Here the sum of angles =$180^0$

- 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

The sum of all the angles around a point is $360^0$

$\angle 1 + \angle 2 + \angle 3 + \angle 4 =360^0$

- 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 - 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 = $\angle 4, \angle 3, \angle 5 , \angle 8$ -
__Corresponding Angles__The angles on the same side of the Transversal are known as Corresponding angles

And Corresponding Angles axiom states that

- Each pair of alternate interior angles are equal

- Each pair of interior angles on the same side of the transversal is supplementary

- Any one pair of corresponding angles are equal

- any one pair of alternate interior angles are equal

- any one pair of interior angles on the same side of the transversal is supplementary

- 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

(a) an acute angled triangle

(b) an obtuse angled triangle

(c) a right triangle

(d) an isosceles 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.

(a) $45^0$

(b) $65^0$

(c) $75^0$

(d) $35^0$

Let x be the two angles equal

then $2x + 50=180$

$x=65$

(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.

(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

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

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 $90^0$

Check your Answer

**Notes**-
**Assignments**

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

- 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.
- 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.
- Pearson IIT Foundation Series - Maths - Class 9 Buy this book if you want to challenge yourself further and want to prepare for JEE foundation.
- 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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