Question

∠g and ∠f are supplementary and ∠h and ∠f are supplementary. complete the proof that ∠g ≅ ∠h.
statement\treason

1. ∠g and ∠f are supplementary\tgiven
2. ∠h and ∠f are supplementary\tgiven
3. m∠g + m∠f = 180°\tdefinition of supplementary angles
4. m∠h + m∠f = 180°\tdefinition of supplementary angles
5. m∠g + m∠f = m∠h + m∠f\t
6. m∠g = m∠h\tproperties of addition, subtraction, multiplication, and divisio
7. ∠g ≅ ∠h\tdefinition of congruence

Explanation:

Step1: State given equal - sums

Since \(m\angle G + m\angle F=180^{\circ}\) and \(m\angle H + m\angle F = 180^{\circ}\), by the transitive property of equality (if \(a = c\) and \(b = c\), then \(a = b\)), we can say \(m\angle G + m\angle F=m\angle H + m\angle F\).

Step2: Subtract \(m\angle F\) from both sides

Using the subtraction property of equality (if \(a + c=b + c\), then \(a=b\)), subtracting \(m\angle F\) from both sides of \(m\angle G + m\angle F=m\angle H + m\angle F\) gives \(m\angle G=m\angle H\).

Step3: 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 have \(\angle G\cong\angle H\).

Answer:

1. Transitive property of equality; 6. Subtraction property of equality