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!!