Question

1. complete the proof by dragging the missing statements and justifications

given: ∠rsu ≅ ∠vst \tprove: ∠rsv ≅ ∠rst
statement\tjustification

statements + justification bank.
definition of congruence\tangle addition postulate\t∠rsu ≅ ∠vst\tm∠rsu + m∠usv = m∠rsv
substitution property\tm∠rsu + m∠usv = m∠rst\tgiven\tdefinition of congruence
m∠rsu = m∠vst\tm∠rsu = m∠vst\ttransitive property\tm∠vst + m∠usv = m∠rst
∠rsv ≅ ∠rst\tangle addition postulate

Explanation:

Step1: Identify Given Information

The given statement is $\angle RSU \cong \angle VST$, so we note this as "Given".

Step2: Apply Angle Addition Postulate

For $\angle RSV$, by Angle Addition Postulate, $m\angle RSU + m\angle USV = m\angle RSV$. Similarly, for $\angle RST$, $m\angle VST + m\angle USV = m\angle RST$.

Step3: Substitute Congruent Angles

Since $\angle RSU \cong \angle VST$, by Definition of Congruence, $m\angle RSU = m\angle VST$.

Step4: Use Substitution Property

Substitute $m\angle RSU$ with $m\angle VST$ in the angle addition equations. So $m\angle VST + m\angle USV = m\angle RSV$ (from first equation substitution) and $m\angle VST + m\angle USV = m\angle RST$ (from second equation). By Transitive Property, $m\angle RSV = m\angle RST$.

Step5: Conclude Congruence

By Definition of Congruence, if $m\angle RSV = m\angle RST$, then $\angle RSV \cong \angle RST$.

Filling the table:

Statement	Justification
$m\angle RSU + m\angle USV = m\angle RSV$	Angle Addition Postulate
$m\angle VST + m\angle USV = m\angle RST$	Angle Addition Postulate
$m\angle RSU = m\angle VST$	Definition of Congruence
$m\angle VST + m\angle USV = m\angle RSV$ (substituted)	Substitution Property
$m\angle RSV = m\angle RST$	Transitive Property
$\angle RSV \cong \angle RST$	Definition of Congruence

Answer:

The completed proof uses the given congruent angles, angle addition postulate, substitution property, transitive property, and definition of congruence to show $\angle RSV \cong \angle RST$. The table is filled as above.