Question

∠e and ∠g are complementary and ∠e and ∠f are complementary. complete the proof that ∠f≅∠g.
statement reason
1 ∠e and ∠g are complementary given
2 ∠e and ∠f are complementary given
3 m∠e + m∠g = 90° definition of complementary angles
4 m∠e + m∠f = 90° definition of complementary angles
5 m∠e + m∠g = m∠e + m∠f transitive property of equality
6 m∠f = m∠g properties of addition, subtraction, multiplication, and division
7 ∠f≅∠g definition of congruence

Explanation:

Step1: State given information

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

Step2: Apply definition of complementary angles

Since $\angle E$ and $\angle G$ are complementary, by the definition of complementary angles, $m\angle E + m\angle G=90^{\circ}$.

Step3: Apply definition of complementary angles again

Since $\angle E$ and $\angle F$ are complementary, by the definition of complementary angles, $m\angle E + m\angle F = 90^{\circ}$.

Step4: Use transitive property of equality

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

Step5: Use properties of equality

Subtract $m\angle E$ from both sides of the equation $m\angle E + m\angle G=m\angle E + m\angle F$. Using the subtraction - property of equality, we get $m\angle F=m\angle G$.

Step6: Apply definition of congruence

Since $m\angle F=m\angle G$, by the definition of congruence (if the measures of two angles are equal, the angles are congruent), $\angle F\cong\angle G$.

Answer:

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