Answer:
Part 1) The value of x is 69°
Part 2) Angle 1=64.5°
Part 3) Angle 2=84.5°
Part 4) Angle 3=31°
Part 5) Angle 4=84.5°
Part 6) Angle 5=95.5°
Part 7) Angle 6=95.5°
Part 8) Angle 7=54.5°
Part 9) Angle 8=30°
Explanation:
Part 1) Find the value of x
we know that
arc SN+arc QS+arc QP+arc PN=360° -----> by complete circle
substitute the values
60°+x°+(x+40)°+(2x-16)°=360°
solve for x
84°+4x°=360°
4x=276°
x=69°
Part 2) Find the measure of angle 1
we know that
The inscribed angle is half that of the arc it comprises
so
m∠1=(1/2)[arc QSN]
arc QSN=arc QS+SN
arc QSN=x+60°=69°+60°=129°
substitute
m∠1=(1/2)[129°]=64.5°
Part 3) Find the measure of angle 2
we know that
The measure of the inner angle is the semi-sum of the arcs that comprise it and its opposite
m∠2=(1/2)[arc SN+arc QP]
substitute the values
m∠2=(1/2)[60°+(x+40)°]
m∠2=(1/2)[60°+(69+40)°]
m∠2=(1/2)[169°]=84.5°
Part 4) Find the measure of angle 3
we know that
The measurement of the outer angle is the semi-difference of the arcs it encompasses.
m∠3=(1/2)[arc PN-arc SN]
substitute the values
m∠3=(1/2)[(2x-16)°-60°]
m∠3=(1/2)[(2(69)-16)°-60°]
m∠3=(1/2)[62°]=31°
Part 5) Find the measure of angle 4
we know that
m∠4=m∠2 -----> by vertical angles
so
m∠4=84.5°
Part 6) Find the measure of angle 5
we know that
m∠5+m∠2=180° -----> by supplementary angles
so
m∠5+84.5°=180°
m∠5=180°-84.5°=95.5°
Part 7) Find the measure of angle 6
we know that
m∠6=m∠5 -----> by vertical angles
so
m∠6=95.5°
Part 8) Find the measure of angle 7
we know that
The inscribed angle is half that of the arc it comprises
so
m∠7=(1/2)[arc QP]
arc QP=(x+40)°=(69+40)°=109°
substitute
m∠7=(1/2)[109°]=54.5°
Part 9) Find the measure of angle 8
we know that
The inscribed angle is half that of the arc it comprises
so
m∠8=(1/2)[arc SN]
arc SN=60°
substitute
m∠8=(1/2)[60°]=30°