Question

Prove that the bisectors of two adjacent supplementary angles include a right angle.

Answer 1

Given:

\(\overrightarrow{CE}\) is the bisector of ∠ACD and

\(\overrightarrow{CF}\) is the bisector of ∠BCD

To Prove: ∠ECF = 90^o

Proof:

From the figure we know that

∠ACD and ∠BCD form a linear pair of angles

So we can write it as

∠ACD + ∠BCD = 180^o

We can also write it as

∠ACE + ∠ECD + ∠DCF + ∠FCB = 180^o

From the figure we also know that

∠ACE = ∠ECD and ∠DCF = ∠FCB

So it can be written as

∠ECD + ∠ECD + ∠DCF + ∠DCF = 180^o

On further calculation we get

2 ∠ECD + 2 ∠DCF = 180^o

Taking out 2 as common we get

2 (∠ECD + ∠DCF) = 180^o

By division we get

(∠ECD + ∠DCF) = 180/2

∠ECD + ∠DCF = 90^o

Therefore, it is proved that ∠ECF = 90^o