Question

in \\( \triangle lmn \\), \\( \overline{ln} \\) is extended through point \\( n \\) to point \\( o \\), \\( m\angle mno = (5x - 13)\degree \\), \\( m\angle nlm = (x - 4)\degree \\), and \\( m\angle lmn = (2x + 19)\degree \\). find \\( m\angle nlm \\).
options:
a \\( 14\degree \\)
b \\( 13\degree \\)
c \\( 11\degree \\)
d \\( 10\degree \\)

Explanation:

Step1: Recall Exterior Angle Theorem

The Exterior Angle Theorem states that the measure of an exterior angle of a triangle is equal to the sum of the measures of the two non - adjacent interior angles. In $\triangle LMN$, $\angle MNO$ is an exterior angle, and the two non - adjacent interior angles are $\angle NLM$ and $\angle LMN$. So, we have the equation: $m\angle MNO=m\angle NLM + m\angle LMN$.

Step2: Substitute the given angle measures

We know that $m\angle MNO=(5x - 13)^{\circ}$, $m\angle NLM=(x - 4)^{\circ}$, and $m\angle LMN=(2x + 19)^{\circ}$. Substituting these into the equation from Step 1, we get:

$$
LATEXBLOCK0
$$

Step3: Solve for x

Subtract $3x$ from both sides of the equation:

$$
LATEXBLOCK1
$$

Add 13 to both sides:

$$
LATEXBLOCK2
$$

Divide both sides by 2:

$$ x=\frac{28}{2}=14 $$

Step4: Find $m\angle NLM$

We know that $m\angle NLM=(x - 4)^{\circ}$. Substitute $x = 14$ into this expression:

$$ m\angle NLM=14 - 4=10^{\circ} $$

Answer:

$\boldsymbol{10^{\circ}}$ (which corresponds to option D)