Question

finding the measure of an arc
in $odot h$, $overparen{ik} cong overparen{jk}$, $moverparen{ik} = (11x + 2)^circ$, and $moverparen{jk} = (12x - 7)^circ$. what is the measure of $overparen{ikj}$?
$moverparen{ikj} = square^circ$

Explanation:

Step1: Set arc measures equal

Since $\widehat{IK} \cong \widehat{JK}$, their measures are equal:
$$11x + 2 = 12x - 7$$

Step2: Solve for $x$

Isolate $x$ by subtracting $11x$ and adding 7:
$$2 + 7 = 12x - 11x$$
$$x = 9$$

Step3: Find $m\widehat{IK}$

Substitute $x=9$ into $\widehat{IK}$'s measure:
$$m\widehat{IK} = 11(9) + 2 = 99 + 2 = 101^\circ$$

Step4: Find $m\widehat{JK}$

Substitute $x=9$ into $\widehat{JK}$'s measure:
$$m\widehat{JK} = 12(9) - 7 = 108 - 7 = 101^\circ$$

Step5: Calculate $m\widehat{IKJ}$

Sum $\widehat{IK}$ and $\widehat{JK}$:
$$m\widehat{IKJ} = 101 + 101 = 202^\circ$$

Answer:

$202$