Question

Write the theorem :- The opposite angles formed by two intersecting lines are of equal measures​

Answer 1

Answer:

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

Proof :

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

Diagram see in the picture.

In the statement above, it is given that ‘two lines intersect each other’. So, let AB and CD be two lines intersecting at O as shown in Fig. 6.8. They lead to two pairs of vertically opposite angles, namely,

(i) ∠ AOC and ∠ BOD (ii) ∠ AOD and ∠ BOC.

We need to prove that ∠ AOC = ∠ BOD and ∠ AOD = ∠ BOC.

Now, ray OA stands on line CD.

Therefore, ∠ AOC + ∠ AOD = 180° (Linear pair axiom) ………..(1)

Can we write ∠ AOD + ∠ BOD = 180°? (Linear pair axiom)……………(2)

From (1) and (2), we can write

∠ AOC + ∠ AOD = ∠ AOD + ∠ BOD

This implies that ∠ AOC = ∠ BOD

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

Answer 2

Answer:

Theorem : the opposite angles formed by intersecting lines are equal measure

Step-by-step explanation:

Given: (i) ∆AOC=∆BOD

(ii) ∆BON=∆AOB

Proof: ∆AOC+∆BOC=180°..........(i) ( angles in linear pair)

∆BOC+∆BON=180°.......(ii) ( angles in linear pair)

∆AOC+∆BOC=∆BON+∆BON......[ FROM (i) and (ii)]

∆AOC=∆BOD....... eliminating∆ BOC.

similarly,it can be proved that ∆BOC=∆AOD

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