Let's solve this step by step to determine which option completes the student's proof correctly.

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Analyzing the Given Information

The proof revolves around the parallelogram $RSTU$, where:

1. $m\angle UTS + m\angle RUT = 180^\circ$ must be proven.
2. The table provided lists 5 steps, with two blanks (Step 2 and Step 4) requiring justification.

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Step-by-Step Analysis

1. Step 1: "Quadrilateral $RSTU$ is a parallelogram."

   • This is given.

2. Step 2:
   We are asked to justify why two lines are parallel.

   • In a parallelogram, opposite sides are parallel. Therefore, $UR \parallel TS$ (correct justification).

3. Step 3: " $m\angle KUR = m\angle UTS$"

   • This follows from the property of corresponding angles between two parallel lines cut by a transversal.

4. Step 4:
   Here, we need to use the fact that two angles form a linear pair.

   • A linear pair means the angles add up to $180^\circ$.
   • Since $m\angle KUR$ and $m\angle RUT$ are on the same line, $m\angle KUR + m\angle RUT = 180^\circ$.

5. Step 5: " $m\angle UTS + m\angle RUT = 180^\circ$"

   • This is justified using the Substitution Property of Equality, combining prior results.

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Correct Steps and Justification

Now, let's compare this analysis with the options provided:

• Option A:

  • Step 2: "Adjacent sides of a parallelogram are parallel" → Incorrect; opposite sides are parallel, not adjacent.
  • Step 4: $m\angle KUR + m\angle RUT = 180^\circ$ → Correct.

• Option B:

  • Step 2: "Adjacent sides of a parallelogram are parallel" → Incorrect for the same reason.
  • Step 4: $m\angle KUR + m\angle URS = 180^\circ$ → Incorrect; $\angle URS$ is unrelated here.

• Option C:

  • Step 2: "Opposite sides of a parallelogram are parallel" → Correct; $UR \parallel TS$.
  • Step 4: $m\angle KUR + m\angle RUT = 180^\circ$ → Correct; this follows from the linear pair.

• Option D:

  • Step 2: "Opposite sides of a parallelogram are parallel" → Correct.
  • Step 4: $m\angle KUR + m\angle URS = 180^\circ$ → Incorrect; this does not align with the linear pair argument.

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Final Answer:

The correct option is C.

• Step 2: Opposite sides of a parallelogram are parallel.
• Step 4: $m\angle KUR + m\angle RUT = 180^\circ$.

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Key Properties Used:

1. Opposite sides of a parallelogram are parallel.
2. Corresponding angles of parallel lines are equal.
3. Linear pairs are supplementary (add up to $180^\circ$).
4. Substitution Property of Equality.

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Do you want me to clarify any part of the solution? Let me know!

Here are 5 related questions to explore further:

1. What are the properties of a parallelogram?
2. How are corresponding angles formed with parallel lines and a transversal?
3. What defines a linear pair of angles?
4. Can you prove that opposite angles in a parallelogram are equal?
5. How do you determine supplementary angles in geometry?

Tip: Always remember that opposite sides and opposite angles in a parallelogram have specific relationships. Keep those properties handy!