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The measure of the arc RQ is 160°

Step-by-step explanation:

Let us use these rules to solve the question

In the given figure

∵ ∠RPQ is an inscribed angle subtended by arc RQ

∴ m of arc RQ = 2 × m∠RPQ

∴ m∠RPQ = 5x + 15

∴ m of arc RQ = 2 × (5x + 15)

→ Multiply the bracket by 2

∴ m of arc RQ = 2(5x) + 2(15)

∴ m of arc RQ = 10x + 30

∵ The measure of the circle = 360°

∵ The sum of the measures of arcs RQ, QP, and PR equals the measure

   of the circle

∵ m of arc QP = 6x - 12

∵ m of arc PR = 9x + 17

→ Add them and equate the sum by 360°

∴ 10x + 30 + 6x - 12 + 9x + 17 = 360

→ Add the like terms in the left side

∵ (10x + 6x + 9x) + (30 - 12 + 17) = 360

∴ 25x + 35 = 360

→ Subtract 35 from both sides

∴ 25x = 325

→ Divide both sides by 25 to find x

∴ x = 13

→ To find the measure of arc RQ substitute x by 13 in its measure

∵ m of arc RQ = 10(13) + 30 = 130 + 30

∴ m of arc RQ = 160°

∴ The measure of the arc RQ is 160°