Question

In the above figure AB and CE are parallel straight lines.
(a) What is the relation between ∠BAC and ∠ACE
(b) Show that, ∠BAC+∠ABC=∠ACD.
(c) Prove that,∠ABC+∠BCE= 2 right angles.

Answer 1

(a) angle BAC and ACE are equal ( alternate interior angles)
(b) We know,
angle BAC = angle ACE ( alt in angles ). ...... eq 1
and,
angle ABC = angle ECD ( co-interior angles) ....... eq 2
Also, we can see,
angle ACE + angle ECD = angle ACD
From eq 1 n 2 we have,
angle BAC + angle ABC = angle ACD (since ACE = BAC and ECD = ABC)
Hence, proved.

(c) We know,
angle BCE + angle ECD = 180 deg (linear pair)
And,
angle ECD = angle ABC ( co-int angles)
Therefore,
angle ABC + angle BCE = 180 deg = 2*90 deg = 2 right angles ( 1 right angle = 90 deg)
angle ABC + angle BCE = 2 right angles.
Hence, proved.

Hope it helps u :)