Question

prove the above axiom​

Answer 1

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Solution

Answer 2

Step-by-step explanation:

l₁ and l₂ are two parallel lines and m is the transversal.

Here, Exterior angles are ∠1, ∠2, ∠7 and ∠8

Interior angles are ∠3, ∠4, ∠5 and ∠6

Corresponding angles are

• ∠1 and ∠5
• ∠2 and ∠6
• ∠4 and ∠8
• ∠3 and ∠7

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

If a transversal intersects two parallel lines then each pair of alternate interior and exterior angles are equal.

(a) ∠1 = ∠7

Also,  (b) ∠1 = ∠3,

Because they are vertically opposite angles.

From (a) and (b),

∠1 = ∠7 and ∠1 = ∠3

⇒∠7  = ∠3

But, ∠7  and  ∠3 corresponding angles.

This can be proved for all sets of corresponding angles.

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