Question

1. if m∠knm=(8x - 5)° and m∠mnj=(4x - 19)°, find the measure of ∠knm.

Explanation:

Step1: Identify angle relationship

$\angle KNM$ and $\angle MNJ$ are vertical - angles, so $m\angle KNM=m\angle MNJ$.

Step2: Set up the equation

$8x - 5=4x-19$.

Step3: Solve for $x$

Subtract $4x$ from both sides: $8x - 4x-5=4x - 4x-19$, which simplifies to $4x-5=-19$. Then add 5 to both sides: $4x-5 + 5=-19 + 5$, so $4x=-14$. Divide both sides by 4: $x=\frac{-14}{4}=-\frac{7}{2}$.

Step4: Find $m\angle KNM$

Substitute $x =-\frac{7}{2}$ into the expression for $m\angle KNM$: $m\angle KNM=8x - 5=8\times(-\frac{7}{2})-5=-28 - 5=-33$. But angle - measure cannot be negative. There is a mistake above. Since $\angle KNM$ and $\angle MNJ$ are vertical angles, we should have $8x - 5=4x + 19$ (assuming a mis - type in the problem setup).

Step5: Correctly solve for $x$

Subtract $4x$ from both sides: $8x-4x - 5=4x-4x + 19$, so $4x-5 = 19$. Add 5 to both sides: $4x-5 + 5=19 + 5$, then $4x=24$. Divide both sides by 4: $x = 6$.

Step6: Calculate $m\angle KNM$

Substitute $x = 6$ into the expression for $m\angle KNM$: $m\angle KNM=8x-5=8\times6-5=48 - 5=43^{\circ}$.

Answer:

$43^{\circ}$