Question

1. if (mangle jkm = 43^{circ}), (mangle mkl=(8x - 20)^{circ}), and (mangle jkl=(10x - 11)^{circ}), find each measure.
2. if (angle def) is a straight - angle, (mangle deg=(23x - 3)^{circ}), and (mangle gef=(12x + 8)^{circ})

Explanation:

Step1: Use angle - addition postulate

Since $\angle{JKL}=\angle{JKM}+\angle{MKL}$, we have the equation $(10x - 11)^{\circ}=43^{\circ}+(8x - 20)^{\circ}$.

Step2: Simplify the right - hand side of the equation

$43+(8x - 20)=8x+43 - 20=8x + 23$. So the equation becomes $10x-11 = 8x + 23$.

Step3: Solve for $x$

Subtract $8x$ from both sides: $10x-8x-11=8x - 8x+23$, which simplifies to $2x-11 = 23$. Then add 11 to both sides: $2x-11 + 11=23 + 11$, so $2x=34$. Divide both sides by 2: $x = 17$.

Step4: Find $m\angle{MKL}$

Substitute $x = 17$ into the expression for $m\angle{MKL}$: $m\angle{MKL}=(8x - 20)^{\circ}=(8\times17-20)^{\circ}=(136 - 20)^{\circ}=116^{\circ}$.

Step5: Find $m\angle{JKL}$

Substitute $x = 17$ into the expression for $m\angle{JKL}$: $m\angle{JKL}=(10x - 11)^{\circ}=(10\times17-11)^{\circ}=(170 - 11)^{\circ}=159^{\circ}$.

Answer:

$x = 17$
$m\angle{MKL}=116^{\circ}$
$m\angle{JKL}=159^{\circ}$