Question

1. in \\( \triangle efg, m \angle f = 3 \cdot m \angle g \\) and the \\( m \angle e = m \angle f - 30 ^ { \circ } \\). find the measure of each angle

Explanation:

Step1: Express angles in terms of \( \angle G \)

Let \( m\angle G=x \). Then \( m\angle F = 3x \), and \( m\angle E=3x - 30^{\circ} \).

Step2: Use triangle - angle sum theorem

By the triangle - angle sum theorem, \( m\angle E+m\angle F+m\angle G = 180^{\circ} \).
Substitute the expressions: \((3x - 30^{\circ})+3x+x=180^{\circ}\).
Combine like terms: \(3x+3x+x-30^{\circ}=180^{\circ}\), so \(7x-30^{\circ}=180^{\circ}\).
Add \(30^{\circ}\) to both sides: \(7x=180^{\circ}+30^{\circ}=210^{\circ}\).
Divide both sides by \(7\): \(x = 30^{\circ}\).

Step3: Find \(m\angle F\) and \(m\angle E\)

Since \(m\angle F = 3x\), then \(m\angle F=3\times30^{\circ}=90^{\circ}\).
Since \(m\angle E=3x - 30^{\circ}\), then \(m\angle E=3\times30^{\circ}-30^{\circ}=60^{\circ}\).

Answer:

\(m\angle E = 60^{\circ}\), \(m\angle F = 90^{\circ}\), \(m\angle G = 30^{\circ}\)