Question

Given ABC with incenter D, find mACD if mACB = 3x + 54 and m ACD = x + 31.

Answer 1

Given:

In triangle ABC, D is incenter m∠ACB = 3x + 54 and m∠ACD = x + 31.

To find:

m∠ACD.

Solution:

We know that,

Incenter of a triangle is the intersection point of all angle bisectors.

`\angle ACD=\angle BCD` ...(i)

Now,

`\angle ACB=\angle ACD+\angle BCD`

`\angle ACB=\angle ACD+\angle ACD` [Using (i)]

`\angle ACB=2\angle ACD`

Substitute the values, we get

`3x+54=2(x+31)`

`3x+54=2x+62`

`3x-2x=62-54`

`x=8`

The value of x is 8.

`m\angle ACD=x+31`

`m\angle ACD=8+31`

`m\angle ACD=39`

Therefore, the measure of angle ACD is 39 degrees.