Question

∠t and ∠u are supplementary and ∠t and ∠v are supplementary. complete the proof that ∠v≅∠u.
statement\treason

1. ∠t and ∠u are supplementary\tgiven
2. ∠t and ∠v are supplementary\tgiven
3. m∠t + m∠u = 180°\t
4. m∠t + m∠v = 180°\t
5. m∠t + m∠u = m∠t + m∠v\t
6. m∠v = m∠u\tproperties of addition, subtraction, multiplication, and division
7. ∠v≅∠u\tdefinition of congruence

Explanation:

Step1: Recall supplementary - angle definition

If two angles are supplementary, the sum of their measures is 180°. So, since $\angle T$ and $\angle U$ are supplementary, $m\angle T + m\angle U=180^{\circ}$ (by the definition of supplementary angles).

Step2: Apply supplementary - angle definition again

Since $\angle T$ and $\angle V$ are supplementary, $m\angle T + m\angle V = 180^{\circ}$ (by the definition of supplementary angles).

Step3: Use the transitive property of equality

Because $m\angle T + m\angle U = 180^{\circ}$ and $m\angle T + m\angle V = 180^{\circ}$, we can say $m\angle T + m\angle U=m\angle T + m\angle V$ (by the transitive property of equality, which states that if $a = c$ and $b = c$, then $a = b$).

Step4: Subtract $m\angle T$ from both sides

Subtracting $m\angle T$ from both sides of the equation $m\angle T + m\angle U=m\angle T + m\angle V$ gives $m\angle V=m\angle U$ (using the subtraction property of equality: if $a + b=a + c$, then $b = c$).

Step5: Use the definition of congruent angles

Since $m\angle V=m\angle U$, by the definition of congruent angles (two angles are congruent if and only if their measures are equal), we have $\angle V\cong\angle U$.

Answer:

1. Definition of supplementary angles
2. Definition of supplementary angles
3. Transitive property of equality
4. Subtraction property of equality