Question

in the diagram below of triangle $bcd$, $e$ is the mid - point of $overline{bd}$ and $f$ is the mid - point of $overline{cd}$. if m$angle dcb=135 - 9x$, and m$angle dfe = 6x + 15$, what is the measure of $angle dfe$?

Explanation:

Step1: Recall mid - segment theorem

Since $E$ is the mid - point of $\overline{BD}$ and $F$ is the mid - point of $\overline{CD}$, by the mid - segment theorem, $EF\parallel BC$.

Step2: Use the property of parallel lines

When two parallel lines are cut by a transversal, corresponding angles are equal. So, $\angle DFE=\angle DCB$.

Step3: Set up the equation

Set $6x + 15=135-9x$.

Step4: Solve the equation for $x$

Add $9x$ to both sides: $6x+9x + 15=135-9x+9x$, which simplifies to $15x + 15=135$.
Subtract 15 from both sides: $15x+15 - 15=135 - 15$, getting $15x=120$.
Divide both sides by 15: $x=\frac{120}{15}=8$.

Step5: Find the measure of $\angle DFE$

Substitute $x = 8$ into the expression for $\angle DFE$: $\text{m}\angle DFE=6x + 15=6\times8+15=48 + 15=75^{\circ}$.

Answer:

$75^{\circ}$