Answer:
Part a) The measure of angle P is [tex]29.9\°[/tex]
Part b) The measure of angle R is [tex]26.3\°[/tex]
Part c) The measure of angle Q is [tex]123.8\°[/tex]
Step-by-step explanation:
step 1
Find the measure of angle P
Applying the law of cosines
[tex]cos(P)=[q^{2}+r^{2}-p^{2}]/[2qr][/tex]
we have
[tex]p=27\ mi[/tex]
[tex]q=45\ mi[/tex]
[tex]r=24\ mi[/tex]
substitute
[tex]cos(P)=[45^{2}+24^{2}-27^{2}]/[2(45)(24)]=0.8667[/tex]
[tex]P=arccos(0.8667)=29.9\°[/tex]
Step 2
Find the measure of angle R
Applying the law of sines
[tex]\frac{p}{sin(P)} =\frac{r}{sin(R)}[/tex]
substitute the values and solve for sin(R)
we have
[tex]p=27\ mi[/tex]
[tex]r=24\ mi[/tex]
[tex]P=29.9\°[/tex]
substitute
[tex]\frac{27}{sin(29.9\°)} =\frac{24}{sin(R)}[/tex]
[tex]sin(R)=sin(29.9\°)*(24)/27=0.4431[/tex]
[tex]R=arcsin(0.4431)=26.3\°[/tex]
step 3
Find the measure of angle Q
we know that
The sum of the interior angles in a triangle must be equal to 180 degrees
so
[tex]m<P+m<Q+m<R=180\°[/tex]
substitute the values
[tex]29.9\°+26.3\°+m<Q=180\°[/tex]
[tex]m<Q=180\°-(29.9\°+26.3\°)=123.8\°[/tex]
