The measure of angle ABC is 101.5 degrees, and the measure of angle CAG is 28 degrees.
If G is the I-center (incenter) of △ABC, this means G is the point where the angle bisectors of the triangle intersect.
The properties of the incenter tell us that the angle bisectors divide the opposite angles into two equal angles.
Hence, if DG is the perpendicular bisector of AB, EG of BC, and GF of AC, then the angles ∠DBG and ∠EBG are bisects of ∠ABC and likewise ∠GCF is a bisect of ∠ACB.
Given that ∠DBG = (3x+4)° and ∠EBG = (5x-8)°, we can set up an equation because these two angles must add up to the full measure of ∠ABC, since they are its bisects.
Therefore the equation is 2*(3x+4) + 2*(5x-8) = 180° (sum of angles in a triangle).
Simplifying gives us 16x - 8 = 180, which results in x = 11.75.
Inserting this value back into the expressions for ∠DBG and ∠EBG, we obtain ∠DBG = 44.25° and ∠EBG = 50.75°.
Hence, the measure of ∠ABC is 2*(5x-8) = 101.5°.
Since ∠GCF = 14° and it is a bisect of ∠ACB, the full measure of ∠CAG is 2*14° = 28°.
The probable question may be:
If g is the I center of ABC In triangle ABC , DG is perpendicular bisector to AB, EG is perpendicular bisector to BC, GF is perpendicular bisector to AC.
angle DBG=(3x+4) degree
angle EBG=(5x-8) degree
angle GCF=14 degree
If g is the center of ABC find each measure
measure of angle ABC=
measure of angle CAG=