Question

for items 3 - 4, use △jkl and △lmn shown. what is m∠lnm?

Explanation:

Step1: Identify vertical angles

Vertical - angles are equal. The angle adjacent to the $72^{\circ}$ angle in $\triangle{JKL}$ and $\angle{MLN}$ are vertical angles. The angle adjacent to the $72^{\circ}$ angle in $\triangle{JKL}$ is $180 - 72=108^{\circ}$. So, $\angle{MLN}=108^{\circ}$.

Step2: Use isosceles - triangle property

In $\triangle{LMN}$, the two sides are equal, so the base - angles are equal. Let $\angle{LNM}=\angle{LMN}=x$.
We know that the sum of the interior angles of a triangle is $180^{\circ}$. So, in $\triangle{LMN}$, we have the equation $x + x+\angle{MLN}=180^{\circ}$.
Substitute $\angle{MLN}=108^{\circ}$ into the equation: $2x+108^{\circ}=180^{\circ}$.

Step3: Solve for $x$

First, subtract $108^{\circ}$ from both sides of the equation: $2x=180^{\circ}-108^{\circ}=72^{\circ}$.
Then, divide both sides by 2: $x = 36^{\circ}$.

Answer:

$36^{\circ}$