Question

given
\\(\overline{ol}\perp\overline{on}\\)
\\(m\angle lom = 3x + 38^{circ}\\)
\\(m\angle mon = 9x + 28^{circ}\\)
find \\(m\angle lom\\):

Explanation:

Step1: Use perpendicular - angle property

Since $\overline{OL}\perp\overline{ON}$, then $\angle LON = 90^{\circ}$, and $\angle LOM+\angle MON=\angle LON = 90^{\circ}$.
So, $(3x + 38)+(9x + 28)=90$.

Step2: Simplify the left - hand side of the equation

Combine like terms: $3x+9x+38 + 28=90$, which gives $12x+66 = 90$.

Step3: Solve for $x$

Subtract 66 from both sides: $12x=90 - 66$, so $12x=24$. Then divide both sides by 12, $x = 2$.

Step4: Find $m\angle LOM$

Substitute $x = 2$ into the expression for $m\angle LOM$. $m\angle LOM=3x + 38=3\times2+38=6 + 38=44^{\circ}$.

Answer:

$44^{\circ}$