Question

1. if ( moverarc{jm} = (10x + 9)^circ ), ( moverarc{nl} = (28x + 17)^circ ), and ( mangle nk l = (16x + 19)^circ ), find ( mangle nk l ).

Explanation:

Step1: Apply intersecting chords angle theorem

The measure of an angle formed by two intersecting chords is half the sum of the measures of the intercepted arcs. So:
$$m\angle NKL = \frac{1}{2}(m\widehat{JM} + m\widehat{NL})$$

Step2: Substitute given expressions

Replace each angle/arc with its algebraic form:
$$16x + 19 = \frac{1}{2}[(10x + 9) + (28x + 17)]$$

Step3: Simplify right-hand side

Combine like terms inside the parentheses first:
$$16x + 19 = \frac{1}{2}(38x + 26)$$
$$16x + 19 = 19x + 13$$

Step4: Solve for $x$

Isolate $x$ by subtracting $16x$ and $13$ from both sides:
$$19 - 13 = 19x - 16x$$
$$6 = 3x$$
$$x = 2$$

Step5: Calculate $m\angle NKL$

Substitute $x=2$ into the expression for $m\angle NKL$:
$$m\angle NKL = 16(2) + 19$$

Answer:

$51^\circ$