Question

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

Explanation:

Step1: Recall supplementary - angle definition

Given that \(\angle G\) and \(\angle F\) are supplementary, so \(m\angle G + m\angle F=180^{\circ}\) (by the definition of supplementary angles). Also, since \(\angle H\) and \(\angle F\) are supplementary, \(m\angle H + m\angle F = 180^{\circ}\) (by the definition of supplementary angles).

Step2: Use substitution property

We have \(m\angle G + m\angle F=m\angle H + m\angle F\) (because both equal \(180^{\circ}\)).

Step3: Apply subtraction property of equality

Subtract \(m\angle F\) from both sides of the equation \(m\angle G + m\angle F=m\angle H + m\angle F\). By the subtraction - property of equality (\(a + c=b + c\) implies \(a = b\)), we get \(m\angle G=m\angle H\).

Step4: Use congruence definition

By the definition of congruent angles (if \(m\angle A=m\angle B\), then \(\angle A\cong\angle B\)), since \(m\angle G=m\angle H\), we can conclude that \(\angle G\cong\angle H\).

Answer:

The proof is completed as shown above to show that \(\angle G\cong\angle H\).