Prove that the bisectors of two adjacent supplementary angles are at right angles

Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts

Dedicated counsellor for each student

Detailed Performance Evaluation

Page 2

Try our Mini CourseMaster Important Topics in 7 DaysLearn from IITians, NITians, Doctors & Academic Experts

Dedicated counsellor for each student

Detailed Performance Evaluation

Question 6 Lines And Angles Exercise 7B

Answer:

To Prove: ∠ECF = 90o

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

We can also write it as

∠ACE + ∠ECD + ∠DCF + ∠FCB = 180

From the figure we also know that

∠ACE = ∠ECD and ∠DCF = ∠FCB

So it can be written as

∠ECD + ∠ECD + ∠DCF + ∠DCF = 180

On further calculation we get

2 ∠ECD + 2 ∠DCF = 180

Taking out 2 as common we get

2 (∠ECD + ∠DCF) = 180

By division we get

(∠ECD + ∠DCF) = 180/2

∠ECD + ∠DCF = 90

Therefore, it is proved that ∠ECF = 90

Video transcript

"hello students welcome to little q a video session i am saf your math tutor and the question for today is prove that the bisector of two adjacent supplementary angles include a right angle so in the figure you can see the two bisectors are given let us see what is given so given to us ce which is the bisector of angle acd and cf which is the bisector of angle bcd we need to prove angle ecf is equal to 90 degree so continuing with the proof it is obvious from the figure that acd angle and bcd angle form a linear pair now as their linear pair we can write angle acd plus angle bcd equals 180 now this equation we can also write it as angle ace plus angle ecd plus angle dcf plus angle fcb which is equal to 180 degree this is just from the figure just by looking at the figure we can see an angle acd is the split of of angle ace and angle ecd in the same way angle bcd is the scripture of angle dcf and angle fcb so we have got this equation now on further calculation from the figure we know that there are two bisectors ec and cf using this bisector property we can say that from figure angle ace is equal to angle ecd and another thing that angle dcf is equal to angle fcb so it can finally be written as we have to substitute these values yeah yeah this values fine so here we can substitute these values as angle ecd plus angle ecd plus angle dcf plus angle dcf should be equal to 180 on further calculation we get 2 times angle ecd plus 2 times angle dcf equals to 180 angle ecd plus angle dcf is equal to 180 divided by two this can be split up like this so one more step i'm adding over here hence we get angle ece angle dcf is equal to 90 therefore angle ecd and angle dcf that sum is nothing but angle ecf so angle ecf is equal to 90 hence proved if you have any questions regarding this you can drop drop it down in our comment section and please subscribe to our lido channel for more such interesting questions thank you "

Answer

Read Less

>

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

Solution

Given: $\angle$ AOC and $\angle$ COB are two supplementary angles. rays OE and OD bisect $\angle$ BOC and $\angle$ AOC respectivey.

To prove : $\angle DOE={90}^o$

Proof: $\angle AOC+\angle COB={180}^o\left(given\right)\therefore \frac{1}{2}\angle AOC+\frac{1}{2}\angle COB=\frac{1}{2}\times {180}^o\therefore \frac{1}{2}\angle AOC+\frac{1}{2}\angle COB={90}^o\Rightarrow \angle DOC+\angle COE={90}^o\left[\text{Since OD bisects}\angle \text{AOC and OE bisects}\angle \text{BOC}\right]\Rightarrow \angle DOE={90}^o$

Mathematics

Secondary School Mathematics IX

Standard IX