Question

1. complete the proof below by proving the missing statements and justifications. given: ∠rsu ≅ ∠vst prove: ∠rsv ≅ ∠ust (there is a diagram with point s and rays sr, su, sv, st. there is a table with columns statement and justification with some green markings on the left side of the table.)

Explanation:

Step1: Identify Given Information

We know that $\angle RSU \cong \angle VST$. Also, $\angle USV$ is a common angle to both $\angle RSV$ and $\angle UST$.

Step2: Apply Angle Addition Postulate

By the Angle Addition Postulate, $\angle RSV = \angle RSU + \angle USV$ and $\angle UST = \angle VST + \angle USV$.

Step3: Substitute Congruent Angles

Since $\angle RSU \cong \angle VST$, we can substitute $\angle RSU$ with $\angle VST$ in the equation for $\angle RSV$. So, $\angle RSV = \angle VST + \angle USV$, which is equal to $\angle UST$ (from the Angle Addition Postulate for $\angle UST$).

Step4: Conclude Congruence

Therefore, by the Substitution Property (or Transitive Property of Congruence for Angles), $\angle RSV \cong \angle UST$.

Answer:

To prove $\angle RSV \cong \angle UST$ given $\angle RSU \cong \angle VST$:

1. Statement: $\angle RSV = \angle RSU + \angle USV$, $\angle UST = \angle VST + \angle USV$

Justification: Angle Addition Postulate

1. Statement: $\angle RSU \cong \angle VST$

Justification: Given

1. Statement: $\angle RSV = \angle VST + \angle USV$

Justification: Substitution (replace $\angle RSU$ with $\angle VST$)

1. Statement: $\angle RSV \cong \angle UST$

Justification: Transitive Property (or Substitution, since $\angle VST + \angle USV = \angle UST$)