Question

suppose that \\( \triangle f g h \\) is isosceles with base \\( \overline{g f} \\).
suppose also that \\( m \angle g=(3 x+47) \\) and \\( m \angle h=(5 x+31) \\).
find the degree measure of each angle in the triangle.

Explanation:

Step1: Use the property of isosceles triangle

Since \(\triangle FGH\) is isosceles with base \(\overline{GF}\), then \(\angle G=\angle F\).
So \(m\angle F=(3x + 47)^{\circ}\).

Step2: Apply the triangle - angle sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\).
So \(m\angle F+m\angle G+m\angle H = 180^{\circ}\).
Substitute \(m\angle F=(3x + 47)^{\circ}\), \(m\angle G=(3x + 47)^{\circ}\), and \(m\angle H=(5x + 31)^{\circ}\) into the equation:
\((3x + 47)+(3x + 47)+(5x + 31)=180\).
Combine like terms:
\(3x+3x + 5x+47+47+31 = 180\).
\(11x+125 = 180\).
Subtract 125 from both sides:
\(11x=180 - 125\).
\(11x=55\).
Divide both sides by 11:
\(x = 5\).

Step3: Calculate the measure of each angle

For \(m\angle F\):
Substitute \(x = 5\) into \(m\angle F=(3x + 47)^{\circ}\).
\(m\angle F=3\times5+47=15 + 47=62^{\circ}\).
For \(m\angle G\):
Since \(m\angle G=m\angle F\), \(m\angle G = 62^{\circ}\).
For \(m\angle H\):
Substitute \(x = 5\) into \(m\angle H=(5x + 31)^{\circ}\).
\(m\angle H=5\times5+31=25+31 = 56^{\circ}\).

Answer:

\(m\angle F = 62^{\circ}\), \(m\angle G = 62^{\circ}\), \(m\angle H = 56^{\circ}\)