Question

in the diagram below of triangle cde, f is the midpoint of (overline{ce}) and g is the midpoint of (overline{de}). if m (angle edc=-2x + 52), and m (angle egf=-9x + 101), what is the measure of (angle egf)?

Explanation:

Step1: Use the mid - segment theorem

Since \(F\) is the midpoint of \(\overline{CE}\) and \(G\) is the midpoint of \(\overline{DE}\), by the mid - segment theorem, \(FG\parallel CD\). When two parallel lines are cut by a transversal (\(DE\) in this case), \(\angle EGF\cong\angle EDC\) (corresponding angles). So, \(-2x + 52=-9x + 101\).

Step2: Solve the equation for \(x\)

Add \(9x\) to both sides: \((-2x+9x)+52=(-9x + 9x)+101\), which simplifies to \(7x+52 = 101\).
Subtract \(52\) from both sides: \(7x+52-52=101 - 52\), so \(7x=49\).
Divide both sides by \(7\): \(x=\frac{49}{7}=7\).

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

Substitute \(x = 7\) into the expression for \(\angle EGF\): \(m\angle EGF=-9x + 101\).
\(m\angle EGF=-9\times7+101=-63 + 101 = 38\).

Answer:

\(38\)