Answer:
m∠P = 61°
m∠Q = 87°
m∠R = 119°
m∠S = 93°
Explanation:
If a quadrilateral is inscribed in a circle, then both pairs of opposite angles are supplementary. Since ∠P and ∠R are opposite angles and therefore supplementary, the sum of their measures is 180°.
m∠P + m∠R = 180°
Substitute the given expressions for the measures of the angles into the equation and solve for x.
6x + 7 + 12x + 11 = 180
Combine like terms.
18x + 18 = 180
Subtract 18 from both sides.
18x = 162
Divide both sides by 18.
x = 9
Substitute this value of x back into the expressions for the angles to find their measures.
m∠P = 6(9) + 7
= 54 + 7
= 61∘
m∠R = 12(9) + 11
= 108 + 11
= 119∘
Since ∠S and ∠Q are also opposite angles and therefore supplementary, the sum of their measures is 180°.
m∠S + m∠Q = 180°
Substitute the given expressions for the measures of the angles into the equation and solve for y.
3(3y + 7) + 10y + 7 = 180
Simplify.
9y + 21 +1 0y + 7 = 180
Combine like terms.
9y + 28 = 180
Subtract 28 from both sides.
19y = 152
Divide both sides by 19.
y = 8
Substitute this value of y back into the expressions for the angles to find their measures.
m∠S = 3(3(8) + 7)
= 3 (24 + 7)
= 3 (31)
= 93°
m∠Q = 10(8 )+7
= 80 + 7
= 87∘
Therefore, m∠P = 61°, m∠Q = 87°, m∠R = 119°, m∠S = 93°.