Question

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

Explanation:

Step1: Identify the mid - segment property

Since \(F\) is the midpoint of \(\overline{CE}\) and \(G\) is the midpoint of \(\overline{DE}\), by the mid - segment theorem of a triangle, \(FG\parallel CD\).
When two parallel lines are cut by a transversal (\(ED\) is the transversal here), \(\angle EGF\) and \(\angle EDC\) are corresponding angles. So, \(\angle EGF=\angle EDC\).

Step2: Set up the equation

Set \(-2x + 52=-9x + 101\).
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 \(\angle EGF\): \(m\angle EGF=-9x + 101\).
\(m\angle EGF=-9\times7+101\).
First, calculate \(-9\times7=-63\). Then, \(-63 + 101=38\).

Answer:

The measure of \(\angle EGF\) is \(38^{\circ}\).