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Identify the angle measures of PQRS.

Identify the angle measures of PQRS.-example-1

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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°.

User Olivier Croisier
by
8.3k points
2 votes
The opposite angles of an inscribe quadrilateral are supplementary.

angle P + angle R = 180
6x + 7 + 12x + 11 = 180
18x + 18 = 180
18x = 162
x = 9

Now we will substitute in the value of x to find angle P and R.

angle P = 6(9)+7 = 61
angle R = 12(9)+11 = 119

Now we will find the value of y.

angle Q + angle S = 180
10y + 7 + 3(3y + 7) = 180
10y + 7 + 9y + 21 = 180
19y + 28 = 180
19y = 152
y = 8

Now substitute in the value of y to find angle Q and S.

angle Q = 10(8)+7 = 87
angle S = 3(3*8+7) = 93

Hope this helps :)
User Xoux
by
7.9k points

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