Question

complete the proof that ∠tvu≅∠rvs.
statement reason
1 m∠rvs + m∠svt = 180° angles forming a linear pair sum to 180°
2 m∠svt + m∠tvu = 180°
3 m∠rvs + m∠svt = m∠svt + m∠tvu properties of addition, subtraction, multiplication, and division
4 m∠rvs = m∠tvu
5 ∠tvu≅∠rvs

Explanation:

Step1: Identify linear - pair property

Angles forming a linear pair sum to 180°. Since $\angle RVS$ and $\angle SVT$ form a linear pair, $m\angle RVS + m\angle SVT=180^{\circ}$. Also, $\angle SVT$ and $\angle TVU$ form a linear pair, so $m\angle SVT + m\angle TVU = 180^{\circ}$ (by the same linear - pair property).

Step2: Use substitution

Because $m\angle RVS + m\angle SVT=180^{\circ}$ and $m\angle SVT + m\angle TVU = 180^{\circ}$, we can substitute the right - hand sides of these equations. So, $m\angle RVS + m\angle SVT=m\angle SVT + m\angle TVU$.

Step3: Apply subtraction property of equality

Subtract $m\angle SVT$ from both sides of the equation $m\angle RVS + m\angle SVT=m\angle SVT + m\angle TVU$. We get $m\angle RVS=m\angle TVU$ (by the subtraction property of equality: if $a + c=b + c$, then $a = b$).

Step4: Define congruent angles

If two angles have equal measures, they are congruent. Since $m\angle RVS=m\angle TVU$, then $\angle TVU\cong\angle RVS$ (by the definition of congruent angles: $\angle A\cong\angle B$ if and only if $m\angle A=m\angle B$).

Answer:

Statement	Reason
2. $m\angle SVT + m\angle TVU = 180^{\circ}$	Angles forming a linear pair sum to $180^{\circ}$
3. $m\angle RVS + m\angle SVT = m\angle SVT + m\angle TVU$	Substitution (from statements 1 and 2)
4. $m\angle RVS = m\angle TVU$	Subtraction property of equality
5. $\angle TVU\cong\angle RVS$	Definition of congruent angles