Question

consider \\( \triangle fgh \\).
complete the following statements.
in \\( \triangle fgh, m \angle f=54^{\circ}, m \angle g= \\) \\( \circ \\), and \\( m \angle h= \\) \\( \circ \\).

Explanation:

Step1: Use the property of isosceles triangle

In \(\triangle FGH\), since \(FG = HG\) (marked by the tick marks), \(\triangle FGH\) is an isosceles triangle. In an isosceles triangle, the angles opposite the equal sides are equal. So, \(m\angle F=m\angle H\). Given \(m\angle F = 54^{\circ}\), then \(m\angle H=54^{\circ}\).

Step2: Use the angle - sum property of a triangle

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \(m\angle G=x\). Then, by the angle - sum property \(m\angle F + m\angle G+m\angle H=180^{\circ}\). Substitute \(m\angle F = 54^{\circ}\) and \(m\angle H = 54^{\circ}\) into the equation: \(54^{\circ}+x + 54^{\circ}=180^{\circ}\).

Step3: Solve for \(m\angle G\)

Simplify the left - hand side of the equation: \(x+108^{\circ}=180^{\circ}\). Subtract \(108^{\circ}\) from both sides: \(x=180^{\circ}- 108^{\circ}=72^{\circ}\).

Answer:

\(m\angle G = 72\), \(m\angle H = 54\)