Question

∠e and ∠g are complementary and ∠e and ∠f are complementary. complete the proof that ∠f≅∠g.
statement\treason
1 ∠e and ∠g are complementary\tgiven
2 ∠e and ∠f are complementary\tgiven
3 m∠e + m∠g = 90°\tdefinition of complementary angles
4 m∠e + m∠f = 90°\tdefinition of complementary angles
5 m∠e + m∠g = m∠e + m∠f\ttransitive property of equality
6 m∠f = m∠g\tproperties of addition, subtraction, multiplication, and division
7 ∠f≅∠g\tdefinition of congruence

Explanation:

Step1: Use given facts

Given $\angle E$ and $\angle G$ are complementary, and $\angle E$ and $\angle F$ are complementary.

Step2: Apply definition of complementary angles

By definition, if two angles are complementary, their sum is $90^{\circ}$. So $m\angle E + m\angle G=90^{\circ}$ and $m\angle E + m\angle F = 90^{\circ}$.

Step3: Use transitive property

Since both $m\angle E + m\angle G$ and $m\angle E + m\angle F$ equal $90^{\circ}$, by the transitive property of equality, $m\angle E + m\angle G=m\angle E + m\angle F$.

Step4: Use subtraction property

Subtract $m\angle E$ from both sides of the equation $m\angle E + m\angle G=m\angle E + m\angle F$. We get $m\angle F=m\angle G$.

Step5: Use definition of congruence

By the definition of congruence, if the measures of two angles are equal, the angles are congruent. So $\angle F\cong\angle G$.

Answer:

The proof is completed as shown above to show $\angle F\cong\angle G$.