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.
[tex]\angle ACD=\angle BCD[/tex] ...(i)
Now,
[tex]\angle ACB=\angle ACD+\angle BCD[/tex]
[tex]\angle ACB=\angle ACD+\angle ACD[/tex] [Using (i)]
[tex]\angle ACB=2\angle ACD[/tex]
Substitute the values, we get
[tex]3x+54=2(x+31)[/tex]
[tex]3x+54=2x+62[/tex]
[tex]3x-2x=62-54[/tex]
[tex]x=8[/tex]
The value of x is 8.
[tex]m\angle ACD=x+31[/tex]
[tex]m\angle ACD=8+31[/tex]
[tex]m\angle ACD=39[/tex]
Therefore, the measure of angle ACD is 39 degrees.
