Class-IX Unit-6 Line and Angles

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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

 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.

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) °

Degree	30°	45°	60°	90°	120°	180°	360°
Radian	π/6	π/4	π/3	π/2	2π/3	π	2π

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

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

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

Theorem Based on Linear Pair Axiom

The sum of all the angles around a point is 360⁰

Vertically Opposite angles

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

Transversal across the parallel Lines

If the transversal intersect two parallel lines as shown in below figure

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

4) Each pair of alternate interior angles are equal

5) 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

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

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