Question

If a transversal intersects two parallel lines then each pair of consecutive interior angles are supplementary prove

Answer 1

Hope this helps you ^_^

Answer 2

Answer:

Given: Lines AB and CD are two parallel lines. Transversal intersects AB at P and CD at Q, making two pairs of consecutive interior angles $\angle$1, $\angle$2 and $\angle$3, $\angle$4

To prove : $\angle$1 + $\angle$2 = 180° and $\angle$3 + $\angle$4 = 180°

Proof : Since ray QD stands on line t

$\therefore$ $\angle$2 + $\angle$5 = 180° [Linear pair]

But, $\angle$1 = $\angle$5. [corresponding angles axiom]

$\therefore$ $\angle$2 + $\angle$1 = 180°

or, $\angle$1 + $\angle$2 = 180° ..... (i)

Now, ray PQ stands on line AB

•°• $\angle$1 + $\angle$3 = 180°.....(ii)

Also, ray QP stands on line CD

•°• $\angle$2 + $\angle$4 = 180° .....(iii)

• Adding (ii) and (iii), we get

$\angle$1 + $\angle$3 + $\angle$2 + $\angle$4 = 180° + 180°

$\rightarrow$ ($\angle$1 + $\angle$2) + $\angle$3 + $\angle$4 = 360°

$\rightarrow$ 180° + $\angle$3 + $\angle$4 = 360°

$\rightarrow$ $\angle$3 + $\angle$4 = 180°

____________

Hence, $\angle$1 + $\angle$2 = 180° and $\angle$3 + $\angle$4 = 180°