Question

prove that the angle between the bisectors of two acute angles of a right triangle is 135 degree.

Answer 1

angel A= 90 degree
angel B= 60 degree
angel C= 30 degree
Now bisecting angel A and angel B
Bisector meet at point D
Then angel D A C= 15degree
Angel D C A=180degree-30degree-45degree=135degree

Answer 2

Given: ABC is the right angle triangle, the right angle will be formed at B.

AO and OC are the angle bisectors of

∠BAC and ∠BCA

To Find That:

∠AOC

Solution:

Since AO and OC are the angle bisectors of ∠BAC and ∠BCA

∠OAC = 1/2 ∠BAC - (1)

∠OCA = 1/2 ∠BCA - (2)

Now we will add equation 1 and 2

∠OAC + ∠OCA = 1/2 ∠BAC + 1/2∠BCA

= 1/2(∠BAC + ∠BCA)

= 1/2 (180-∠ABC)

(the sum of interior angles is 180 degrees)

∠OAC + ∠OCA = 1/2 [180-90]

= 1/2 * 90

= 45

- (3)

Now in the triangle AOC,

∠AOC = 180 - [∠OAC + ∠OCA]

= 180 - 45

= 135

So the angle at O between the two bisectors is 135 Degrees.