Question

in the diagram below of triangle $abc$, $d$ is the mid - point of $overline{ac}$ and $e$ is the mid - point of $overline{bc}$. if m$angle cba=5x + 48$, and m$angle ced=87 - 8x$, what is the measure of $angle cba$?

Explanation:

Step1: Apply mid - point theorem property

Since $D$ is the mid - point of $\overline{AC}$ and $E$ is the mid - point of $\overline{BC}$, by the mid - point theorem, $DE\parallel AB$.

Step2: Use the property of corresponding angles

When $DE\parallel AB$, $\angle CED$ and $\angle CBA$ are corresponding angles, so $\angle CED=\angle CBA$.

Step3: Set up the equation

Set $5x + 48=87 - 8x$.

Step4: Solve the equation for $x$

Add $8x$ to both sides: $5x+8x + 48=87-8x + 8x$, which gives $13x+48 = 87$. Then subtract 48 from both sides: $13x+48 - 48=87 - 48$, so $13x=39$. Divide both sides by 13: $x = 3$.

Step5: Find the measure of $\angle CBA$

Substitute $x = 3$ into the expression for $\angle CBA$: $\text{m}\angle CBA=5x + 48=5\times3+48=15 + 48=73^{\circ}$.

Answer:

$73^{\circ}$