LINES AND ANGLES (LECTURE 3)

BLOG - LINES & ANGLES - 28TH APRIL

GOOD MORNING BOYS ...

TODAY WE START WITH  NCERT Ex 6.1

KINDLY NOTE DOWN ALL THE ANSWERS IN YOUR NOTEBOOK

KINDLY NOTE , WHENEVER THE ANGLES ARE GIVEN IN A RATIO LIKE 2 :3 we assume 

the angles to be 2x and 3x where " x" is the common factor 

NOTE THAT SUM OF ALL THE ANGLES ON A STRAIGHT LINE IS 180 degrees

LINES AND ANGLES ( LECTURE 2)

LINES AND ANGLES 2

DEAR STUDENTS

WELCOME TO THE CLASS!!

FEW REMINDERS:

The text in Red, is to be written in your register

·         The text in blue is to be viewed by clicking on it

·         The text in green is to be practiced for home work

1.      take your  Mathematics Register

2.      Our handwriting reflects a lot about us. It will be awesome if you use good presentation and cursive hand writing

3.      Make a column on the right hand side, if you need to do any rough work

4.      Leave two lines where you finished yesterday’s work and draw a horizontal line

5.      Write today's date on the line after that.

6.      Pending Queries from yesterday (if any).......

Yesterday's  learning outcome 

1. recall,recognize and define lines , line segment and ray

2 recall,identify and define obtuse,acute and right angle.

3. recall . identify and define pair of angles

Today we will cover the followinglearning outcomes:

I will be able to:

1. recall and define parallel and intersecting lines.

2. state and apply linear pair axiom

3. prove If two lines intersect each other, then the vertically opposite angles are equal.

Draw two different lines PQ and RS on a paper.
In how many ways  you can draw them ?
The answer is just 2 ways.

WRITE DOWN AND DRAW  FIGURES

Let's us discuss a very important Axiom

DRAW THE FIGURE GIVEN BELOW

Here, we are given a line AB and a ray OC standing on it

Can we write ∠ AOC + ∠ BOC = ∠ AOB? 

What is the measure of ∠ AOB? It is 180°. (Why?) 

 can you say that ∠ AOC + ∠ BOC = 180°? Yes! 

So, we can conclude that: If a ray stands on a line, then the sum of two adjacent angles so formed is 180°......(A)

Now, look at the  adjacent angles of different measures as shown BELOW. Keep the ruler along one of the non-common arms in each case.

 Does the other non-common arm also lie along the ruler?

From the above discussion, we can state that:

If the sum of two adjacent angles is 180°, then the non-common arms of the angles form a line.....(B)

Statements (A) and (B) together known as linear pair AXIOM

They are converse of each other

Click on the link for more clarification

NOTE DOWN

LINEAR PAIR AXIOM

LET US PROVE A VERY IMPORTANT PROPERTY ABOUT VERTICALLY OPPOSITE ANGLES

WE HAVE ALREADY STATED IT IN THE PREVIOUS CLASS!!

CLICK ON THE LINK TO SEE THE PROOF and 2 EG. BASED ON THE RESULTS DONE

Thm 6.1: If two lines intersect each other, then the vertically opposite angles are equal.

(WE ALL KNOW THIS BUT NOW WE WILL DO ITS PROOF)

(PLZ NOTE IT DOWN)

Given: AB and CD be two lines intersecting at O as 
To prove: ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.

Proof:Now, ray OA stands on line CD.
Therefore, ∠ AOC + ∠ AOD = 180° (Linear pair axiom) (1)
Similarly, ∠ AOD + ∠ BOD = 180°  (2)
From (1) and (2), we can write
                                  ∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD(using Axiom: things equal to same thing are equal to one another)

This implies that ∠ AOC = ∠ BOD (subtracting equals (∠ AOD )from each side)

Similarly, it can be proved that ∠AOD = ∠BOC


LET'S DO SOME SOLVED EXAMPLES BASED ON THE RESULTS DONE SO FAR

Solved eg 1. and 2 , were discussed in the link above

You can note them down from page 95 and 96 (ncert book) as HW

Now. look at the solved eg 3

NOTE THIS DOWN
Example 3 : In Fig. 6.11, OP, OQ, OR and OS are four rays. Prove that ∠ POQ + ∠ QOR + ∠ SOR + ∠ POS = 360°.

Now, we will produce  ray  backwards to a point T

.

Now, ray OP stands on line TOQ.
Therefore, ∠ TOP + ∠ POQ = 180° (1)
(Linear pair axiom)

Similarly, ray OS stands on line TOQ.
Therefore, ∠ TOS + ∠ SOQ = ........................(2)
But ∠ SOQ = ∠ SOR + ∠ QOR (in the figure ∠ SOR and ∠ QOR combined together make ∠ SOQ)
So, (2) becomes
∠ TOS + ∠ SOR + ∠ QOR = 180° (3)
Now, adding (1) and (3), you get
∠ TOP + ∠ POQ + ∠ TOS + ∠ SOR + ∠ QOR = 360° (4)
But ∠ TOP + ∠ TOS = ∠ POS
Therefore, (4) becomes
∠ POQ + ∠ QOR + ∠ SOR + ∠ POS = 360°

Go through the next SDG goal

We end our class here!!
Keep revising!!
It is very important!!

LINES AND ANGLES (LECTURE 1)

Lines and angles 1
DEAR STUDENTS

WELCOME TO THE CLASS!!

FOR MARKING YOUR ATTENDANCE↔☟

FILL THIS FORM


FEW REMINDERS:

The text in Red, is to be written in your register

·         The text in blue is to be viewed by clicking on it

·         The text in green is to be practiced for home work

1.      take your  Mathematics Register

2.      Our handwriting reflects a lot about us. It will be awesome if you use good presentation and cursive hand writing

3.      Make a column on the right hand side, if you need to do any rough work

4.      Leave two lines where you finished yesterday’s work and draw a horizontal line

5.      Write today's date on the line after that.

6.      Pending Queries from yesterday (if any).......

7.      Please write the learning outcomes as mentioned below

Today's learning outcome is 

I will be able to:
1. recall,recognize and define lines , line segment and ray
2 recall,identify and define obtuse,acute and right angle.
3. recall . identify and define pair of angles




You can see plenty of examples around where lines and angles are used. In the chapter" LINES AND ANGLES" you will study about their properties and its applications.

↔⭐All figures to be drawn neatly using pencil and scale

Look at the image →☟
Select the correct option

let's discuss the answers(NOTE DOWN ALL DEFINITIONS ) 
(DRAW FIG 1,2 AND 3)
1. Line
🔴 Aline is one dimensional straight figure which can be extended indefinitely in both the directions.

When we draw lines in geometry, we use an arrow at each end to show that it extends infinitely. 

A line can be named either using two points on the line (for example, AB←→ ) or simply by a letter, usually lowercase (for example, line 
$\mathrm{<b>m</b>}$ ).

2.Line segment
🔴A line segment has two endpoints.  You can measure the length of a segment, but not of a line.

A segment is named by its two endpoints, 

for example,

3.Ray KY
🔴 A ray is a part of a line that has one endpoint and goes on infinitely in only one direction. You cannot measure the length of a ray. 
A ray is named using its endpoint first, and then any other point on the ray

4.Line segment

    5.option 1


   6. line segments

All of you draw 2 points A and B in your notebook

  If you join them what do you get? 
Obviously a straight line.



Now take 3 points as shown below

.A       .B               .C

If you join all 3 points then again you get a straight line.

Look at the image ☟

If you  join all 3 points then do you get a straight line?
The answer is NO

Such points are called as non collinear points

COLLINEAR POINTS(NOTE  DOWN)
If three or more points lie on the same line, they are called collinear points; otherwise they are called non-collinear points.


In your daily life, you see different types of angles formed

What are the different types of angles?(draw them neatly )

Let's discuss about different pairs of angles
Click on the link 

Identify the type of pair of angles in the image below

Let's discuss the answers
(WRITE DOWN ALL DEFINITIONS AND COPY THE CORRESPONDING FIGURES GIVEN ABOVE)

1.  Two angles whose sum is 90° are called complementary angles, 


2.Two angles whose sum is 180° are called supplementary angles.


3. Two angles are adjacent, if they have a common vertex, a common arm and their non-common arms are on different sides of the common arm

4. The vertically opposite angles formed when two lines, say AB and CD, intersect each other, say at the point O . There are two pairs of vertically opposite angles.

a) ∠1 = ∠2 ; ∠3 = ∠4
b) one pair is acute and other is obtuse


5.If the non-common arms  in Fig. 3, form a line then it will look like Fig.5. In this case, angles 1 and 2 are called linear pair of angles
Are these angles supplementary and adjacent also?
So, we can say
ANGLES FORMING LINEAR PAIR WILL ALWAYS BE SUPPLEMENTARY

BUT IT IS NOT TRUE VICE VERSA

EG. IS FIG 2 ABOVE
You can utilize your double period to finish this work, use ruler and pencil to draw neat figures in your register. Next class on wards keep your geometry box ready.
NOTE DOWN

THE CLASS ENDS HERE TODAY!

REVISE CW !!
SEE YOU FOR THE NEXT CLASS!!