Question

an isosceles triangle is shown. what is the measure of \\( \angle f \\)?
a \\( 32 ^ { \circ } \\)
b \\( 58 ^ { \circ } \\)
c \\( 64 ^ { \circ } \\)
d cannot be determined

Explanation:

Step1: Use the property of isosceles triangle

In an isosceles triangle \( \triangle FGM\), \(FG = MG\), so \( \angle F=\angle M\).

Step2: Apply the triangle - angle - sum theorem

The sum of the interior angles of a triangle is \(180^{\circ}\). Let \( \angle F = x\), then \( \angle M=x\), and \( \angle G = 58^{\circ}\).
We have the equation \(x + x+58^{\circ}=180^{\circ}\).
Simplify the left - hand side: \(2x+58^{\circ}=180^{\circ}\).
Subtract \(58^{\circ}\) from both sides: \(2x=180^{\circ}-58^{\circ}=122^{\circ}\).
Divide both sides by 2: \(x = 61^{\circ}\). Wait, no, there is a mistake. Wait, no, actually, if \(FG = MG\), then \( \angle F=\angle M\). Using the angle - sum formula \( \angle F+\angle M+\angle G=180^{\circ}\). Since \( \angle F=\angle M\), we have \(2\angle F + 58^{\circ}=180^{\circ}\). Then \(2\angle F=180^{\circ}-58^{\circ}=122^{\circ}\), \( \angle F = 61^{\circ}\). No, wait, no, the problem may have a mis - labeling. Wait, if it is \( \triangle FGM\) with \(FG = GM\) (the marks on the sides), then \( \angle F=\angle G\) is wrong. Wait, no, the side - mark: if two sides are equal (the marks on \(FG\) and \(MG\)), then the base angles are \( \angle M\) and \( \angle F\). The formula \( \angle F+\angle M+\angle G = 180^{\circ}\), and \( \angle F=\angle M\). So \(2\angle F+58^{\circ}=180^{\circ}\), \(2\angle F=122^{\circ}\), \( \angle F = 61^{\circ}\). But this is not in the options. Wait, maybe the side - mark is \(FG = FM\). If \(FG = FM\), then \( \angle G=\angle M\). Then \( \angle F+2\angle G=180^{\circ}\). If \( \angle G = 58^{\circ}\), then \( \angle F=180^{\circ}-2\times58^{\circ}=180^{\circ}-116^{\circ}=64^{\circ}\).

Answer:

C. \(64^{\circ}\)