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

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 chapterJust a recap to help you

a)

b)

c)

d)

e)

We denote the angle by symbol ∠

If A is the Vertex,then angle could be represented as ∠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 |

∠A=30

∠B=30

then

∠A=∠B

They are congurent

Here the sum of angles =90

Here the sum of angles =180

And If the sum of the adjacent angles is 180

The sum of all the angles around a point is 360

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

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

a) The sum of the angles of the triangle is 180

b) if the side of the triangle is produced ,the exterior angle formed is equal to the sum of the opposite interior angle

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.

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

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 haveall 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

(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

15x = 180

x = 12

Hence angles 60

2) An exterior angle of a triangle is 130

(a) 45

(b) 65

(c) 75

(d) 35

Let x be the two angles equal

then 2x + 50=180

x=65

3)

(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

(e) A triangle can have all angles more than 60

(f) A triangle can haveall angles equal to 60

g) The two acute angles in every right triangle are complementary.

(a) False. Since the sum of three angles of a triangle is 180

(b) True Because the sum of two obtuse angles will become greater than 180

(c) True a triangle can have two acute angles. Since the sum of two acute angle is less than 180

(d) False Since sum of all angles less than 60

(e) False Since sum of angles (each angle is greater than 60

(f) True Sum of angles (each = 60

g) True

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