Question

given g is the incenter of triangle abc choose the correct lengths and angle measurements.
m∠gcf = (blank 1)
m∠cgf = (blank 2)
m∠dag = (blank 3)
m∠bgd = (blank 4)
m∠dbg = (blank 5)
m∠bge = (blank 6)
da ≈ (blank 7)

Explanation:

Step1: Find $\angle GCF$

$m\angle GCF = 31^\circ$ (G is incenter, bisects $\angle C$)

Step2: Find $\angle CGF$

Right triangle $CFG$: $\angle CFG=90^\circ$
$m\angle CGF = 90^\circ - 31^\circ = 59^\circ$

Step3: Find $\angle DAG$

First calculate $\angle BAC$:
Sum of angles in $\triangle ABC$: $180^\circ - 2\times21^\circ - 2\times31^\circ = 180^\circ-42^\circ-62^\circ=76^\circ$
$m\angle DAG = \frac{76^\circ}{2}=38^\circ$ (G is incenter, bisects $\angle A$)

Step4: Find $\angle BGD$

Right triangle $BDG$: $\angle BDG=90^\circ$, $\angle DBG=21^\circ$
$m\angle BGD = 90^\circ - 21^\circ = 69^\circ$

Step5: Find $\angle DBG$

$m\angle DBG = 21^\circ$ (G is incenter, bisects $\angle B$)

Step6: Find $\angle BGE$

Right triangle $BEG$: $\angle BEG=90^\circ$, $\angle EBG=21^\circ$
$m\angle BGE = 90^\circ - 21^\circ = 69^\circ$

Step7: Find $DA$

$AB = 12.3$, $BD=4.2$, so $DA = AB - BD = 12.3 - 4.2 = 8.1$

Answer:

$m\angle GCF = 31^\circ$
$m\angle CGF = 59^\circ$
$m\angle DAG = 38^\circ$
$m\angle BGD = 69^\circ$
$m\angle DBG = 21^\circ$
$m\angle BGE = 69^\circ$
$DA \approx 8.1$