Question

in the diagram below of triangle abc, d is the mid - point of ac and e is the mid - point of bc. if m∠cab = 38 + 6x, and m∠cde = 64 - 7x, what is the measure of ∠cde?

Explanation:

Step1: Use the mid - point theorem

Since \(D\) is the mid - point of \(\overline{AC}\) and \(E\) is the mid - point of \(\overline{BC}\), by the mid - point theorem, \(DE\parallel AB\). Then \(\angle CDE=\angle CAB\) (corresponding angles).

Step2: Set up the equation

Set \(38 + 6x=64 - 7x\).
Add \(7x\) to both sides: \(38+6x + 7x=64-7x + 7x\), which simplifies to \(38 + 13x=64\).
Subtract 38 from both sides: \(13x=64 - 38\), so \(13x=26\).
Divide both sides by 13: \(x = 2\).

Step3: Find the measure of \(\angle CDE\)

Substitute \(x = 2\) into the expression for \(\angle CDE\).
\(\angle CDE=64-7x\), so \(\angle CDE=64-7\times2=64 - 14 = 50^{\circ}\).

Answer:

\(50^{\circ}\)