Question

find the measure of ∠gmj. m∠gmj = °

Explanation:

Step1: Use vertical - angle property

Vertical angles are equal. $\angle GMJ$ and $\angle KNL$ are vertical angles, so $12x - 1=13x - 44$.

Step2: Solve for $x$

Subtract $12x$ from both sides: $- 1=x - 44$. Then add 44 to both sides, we get $x = 43$.

Step3: Find the measure of $\angle GMJ$

Substitute $x = 43$ into the expression for $\angle GMJ$: $m\angle GMJ=12x - 1=12\times43-1$.
$12\times43-1 = 516 - 1=515$. But angles should be between $0^{\circ}$ and $180^{\circ}$, we made a mistake. $\angle GMJ$ and $\angle KNL$ are corresponding angles. Since $IJ\parallel KL$, we have $12x - 1+13x - 44 = 180$ (same - side interior angles are supplementary).

Step4: Combine like terms

$(12x+13x)+(-1 - 44)=180$, so $25x-45 = 180$.

Step5: Solve for $x$

Add 45 to both sides: $25x=180 + 45=225$. Then divide both sides by 25, $x = 9$.

Step6: Find the measure of $\angle GMJ$

Substitute $x = 9$ into the expression for $\angle GMJ$: $m\angle GMJ=12x - 1=12\times9-1=108 - 1 = 107$.

Answer:

$107$