Question

∠e and ∠g are supplementary and ∠f and ∠e are supplementary. complete the proof that ∠g ≅ ∠f.
1 ∠e and ∠g are supplementary given
2 ∠f and ∠e are supplementary given
3 m∠e + m∠g = 180° definition of supplementary angles
4 m∠f + m∠e = 180°
5 m∠e + m∠g = m∠f + m∠e
6 m∠g = m∠f properties of addition, subtraction, multiplication, and division
7 ∠g ≅ ∠f

Explanation:

Step1: Definition of supplementary angles

$$m\angle F + m\angle E=180^{\circ}$$

Step2: Substitution property of equality

Since \(m\angle E + m\angle G = 180^{\circ}\) (from step 3) and \(m\angle F + m\angle E = 180^{\circ}\) (from step 4), we can substitute \(180^{\circ}\) in the first equation with \(m\angle F + m\angle E\). So, \(m\angle E + m\angle G=m\angle F + m\angle E\)

Step3: Definition of congruent angles

If \(m\angle G=m\angle F\), then by the definition of congruent angles (two angles are congruent if their measures are equal), \(\angle G\cong\angle F\)

Answer:

1. Definition of supplementary angles; 5. Substitution property of equality; 7. Definition of congruent angles