Lines and Angles: 9 Math


mathematics Class Nine

Basic Terms And Definitions: 9th math

Line Segment

A part of a line with two end points is called a Line Segment.

class 9th math lines and angles line segment

A line segment is denoted by `bar(AB)`

Ray

A part of a line with one end point is called a Ray.

class 9th math lines and angles a ray

A ray is denoted by `vec(AB)`.

Collinear Points and Non-collinear Points

If three or more points lie on the same line, they are called Collinear Points. Otherwise they are called Non-Collinear Points

class 9th math lines and angles collinear and non collinear points

Example A, B, C and D are collinear points.

While A, B and E are non collinear points.

Angle

When two rays originate from the same end point an angle is formed.

Example: `vecOA` and `vecOB` are two rays meeting at a point O. Thus, these rays form an angle called `/_BOA` and is read as angle BOA.

class 9th math lines and angles an angle

Here, in the example OB and OA are arms, and O is the vertex.

Arms and Vertex

The rays making an angle are called Arms while the point where both the arms meet is called the Vertex.

Here OB and OA are arms of the angle and O is called the vertex of the angle.

Types of Angle

On the basis of measurement angles can be divided into five types. These are Acute angle, Right angle, Obtuse angle, Straight angle and Reflex angle.

Acute Angle

Angle greater than 0 and less than 90o is called an Acute Angle.

class 9th math lines and angles acute angle

Acute Angle

Here, Angle is 60o. Since it is less than 90o and greater than 0o, thus it is an example of Acute Angle.

Similarly, angles of 10o, 15o, 20o, 70o, etc. are some other examples of Acute Angles.

(ii) Right Angle

Angle equal to 90o is called a Right Angle.

class 9th math lines and angles a right angle

Since the measurement of the given angle is equal to 90o, thus it is a Right Angle.

(iii) Obtuse Angle

Angle greater than 90o and less than 180o is called an Obtuse Angle.

class 9th math lines and angles obtuse angle

Here, since the given angle is 120o which is greater than 90o, and less than 180o, thus it is an example of Obtuse Angle.

(iv) Straight Angle

Angle equal to 180o is called a Straight Angle

class 9th math lines and angles straight angle

Here since the measurement of the given angle is equal to 180o, thus it is a Straight Angle.

(v) Reflex Angle

Angle greater than 180o and less than 360o is called a Reflex Angle.

class 9th math lines and angles reflex angle

Here it is an angle of 325o. Since the measurement of this angle is greater than 180o and less than 360o, thus it is an example of Obtuse Angle.

Similarly, angles of 200o, 240o, 310o, 325o, etc. are examples of Obtuse Angles.

Pairs of Angles

Complementary Angles

If the sum of two angles is equal to 90o, then angles are called Complimentary Angles.

class 9th math lines and angles complementary angles

Example

Here, `/_AOB` and `/_BOC` are complementary angles because their sum (30o + 60o = 90o) is equal to 90o.

Supplementary Angles

If the sum of two angles is equal to 180o, then angles are called Supplementary Angles.

class 9th math lines and angles supplementary angles

Example

Adjacent Angles

If two angles have a common vertex and a common arm and their non-common arms are on different sides of the common arm, angles are called Adjacent Angles.

class 9th math lines and angles adjacent angles

Example

Here, angle AOB and angle BOC are adjacent angles. Because ray OB and vertex O is common in both the angles.

Linear Pair of Angles

If non-common arms of adjacent angles form a straight line, then angles are called Linear Pair of Angles.

class 9th math lines and angles linear pair of angles

Example

Here, non-common arms OA and OB of angle AOC and COB form a straight line, thus, angle AOC and angle COB are linear pair of angles.

Vertically Opposite Angles

If two straight lines intersect each other, the at the point of intersection four angles are formed. The opposite angles are called Vertically Opposite Angles. There are two pairs of vertically opposite angles are formed at a point of intersection of two straight line.

class 9th math lines and angles vertically opposite angles

The vertically opposite angles are equal in measurement.

Here, angle 1 and angle 3 are vertically opposite angles. Similarly angle 2 and angle 4 are vertically opposite angles.

And angle 1 = angle 3.

And angle 2 = angle 4.

Parallel Lines and Transversal

class 9th math lines and angles parallel lines and transversal

Let there are two parallel lines m and n.

And a transversal t intersects these two parallel lines at distinct point.

Let angles made by these parallel lines and transversal are 1, 2, 3, 4, 5, 6, 7 and 8

In such condition, ∠1, ∠2, ∠7 and ∠8 are called Exterior Angles .

And ∠3, ∠4, ∠5 and ∠6 are called Interior Angles.

(a) Corresponding Angles

(i) ∠1 and ∠5, (ii) ∠2 and ∠6, (iii) ∠4 and ∠8 and (iv) ∠3 and ∠7

(b) Alternate Interior Angles

(i) ∠4 and ∠6 (ii) ∠3 and ∠5

(c) Alternate Exterior Angles

(i) ∠1 and ∠7 (ii) ∠2 and ∠8

(d) Interior Angles on the same side of the Transversal

(i) ∠4 and ∠5 (ii) ∠3 and ∠6

Linear Pair Axioms

Axiom 1

If a ray stands on a line, then the sum of two adjacent angles so formed is 180o.

Axiom 2

If the sum of two adjacent angles is 180o, then the non common arms of the angles forma a line.

Axiom 2 is called the Converse of Axiom 1.

Since these two axioms says about linear pair of angle, thus, these are known as Linear Pair Axiom.

Axion 3

If a transversal intersects two parallel lines, then each pair of corresponding angles is equal.

Axiom 3 is called the corresponding angles axiom.

Axiom 4

If a transversal intersects two lines such that a pair of corresponding angles is equal, then the two lines are parallel to each other.

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