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Class 9 Maths notes for Geometry



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 alpabet. 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 infinate 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 lenght 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) Congurent segments: If two line segments are equal then they are called congurent
Now we will extend our knowlegde to angles in this chapter

What is angle

An angle is a formed of 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 ∠
If A is the Vertex,then angle could be represented as ∠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 center 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
∠A=300
∠B=300
then
∠A=∠B
They are congurent

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
Adjacent angle

Complimentary Angles

Two angles , the sum of whose measures is 900 is called Complimentary Angles.Each of these Complimentary Angles are called the complement of each other
Complimentary Angles
Here the sum of angles =900

Supplementary Angles

Two angles , the sum of whose measures is 1800 is called Supplementary Angles.Each of these Supplementary Angles are called the suplement of each other
Supplementary Angles
Here the sum of angles =1800

Linear Pair Axioms

If a ray stands on a line, then the sum of the adjacent angles so formed is 1800
And If the sum of the adjacent angles is 1800,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 3600
he sum of all the angles around a point

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 aways from the figure
1) We can see following angles as depicted in the figure above
∠1,∠2,∠3,∠4 on the first parallel line
and ∠5,∠6,∠7,∠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
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 supplimentary
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
a) any one pair of corresponding angles are equal
b) any one pair of alternate interior angles are equal
c) any one pair of interior angles on the same side of the transversal is supplimentary
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
a) The sum of the angles of the triangle is 1800
b) 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

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 = 1800
15x = 1800
x = 120
Hence angles 600, 360, 840 .SO it is an acute angle triangle.

2) An exterior angle of a triangle is 1300 and its two interior opposite angles are equal. Each of the interior angle is equal to:
(a) 450
(b) 650
(c) 750
(d) 350
Solution
Let x be the two angles equal
then 2x + 50=180
x=65

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 600
(e) A triangle can have all angles more than 600
(f) A triangle can haveall angles equal to 600
g) The two acute angles in every right triangle are complementary.

Solution
(a) False. Since the sum of three angles of a triangle is 1800. 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 1800. 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 1800, the third angle will have the remaining value.
(d) False Since sum of all angles less than 600 is still less than 1800. Such a triangle is not possible.
(e) False Since sum of angles (each angle is greater than 600) exceed 1800, such a triangle is not possible.
(f) True Sum of angles (each = 600) is exactly equal to 1800, such a triangle is possible. It makes an equilateral triangle.
g) True


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