Essay
Your teacher will grade your response for the following question to ensure that you receive proper credit for your answer.
Given: Triangle SVX is congruent to triangle UTX and line segment SV is parallel to line segment TU. Prove: VUTS is a parallelogram.
The image of a quadrilateral is shown. Its four vertices are labeled V, U, T, and S. One diagonal is drawn from vertex S to vertex U. Another diagonal is drawn from vertex T to vertex V. The two diagonals intersect in the middle of the quadrilateral at a point labeled X.
Complete the two-column proof.
A two-column proof is shown with four statements and four corresponding reasons. The four statements are in the left column and are labeled 1 through 4. Statement 1 says triangle SVX is congruent to triangle UTX. Statement 2 is blank. Statement 3 is blank. Statement 4 stays VUTS is a parallelogram. The four reasons are in the right column and are labeled 5 through 8. Statement 5 says given. Statement 6 says given. Statement 7 is blank. Statement 8 is blank.• In the first column under Statement are the following statements from top to bottom:
1. triangle upper S upper V upper X congruent to triangle upper U upper T upper X
2. modifying below blank with bar
3. modifying below blank with bar
4. upper V upper U upper T upper S is a parallelogram period
• In the second column under Reason are the following statements from top to bottom:
5. Given
6. Given
7. modifying below blank with bar
8. modifying below blank with bar
(4 points)
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To prove that VUTS is a parallelogram, we can use the given information that triangle SVX is congruent to triangle UTX and that line segment SV is parallel to line segment TU.
Here is the completed two-column proof:
Statement
1. triangle SVX is congruent to triangle UTX
2. SV is parallel to TU
3. UV ≅ TS, SX ≅ XV (corresponding parts of congruent triangles are congruent)
4. VUTS is a parallelogram
Reason
5. Given
6. Given
7. Definition of congruent triangles (corresponding parts are congruent)
8. Definition of a parallelogram (opposite sides are congruent)
By using the given congruence and the fact that SV is parallel to TU, we can conclude that VUTS is indeed a parallelogram.
To prove that VUTS is a parallelogram, we can use the given congruence of triangle SVX and triangle UTX, as well as the fact that line segment SV is parallel to line segment TU.
Here is a complete two-column proof:
1. triangle SVX is congruent to triangle UTX 5. Given
2. line segment SV is parallel to line segment TU 6. Given
3. angle SVX is congruent to angle UTX 7. Corresponding parts of congruent triangles are congruent (CPCTC)
4. angle VUT is congruent to angle STU 8. Alternate interior angles are congruent (theorem for parallel lines)
5. line segment SV is congruent to line segment TU 7. Corresponding parts of congruent triangles are congruent (CPCTC)
6. line segment VT is congruent to line segment US 8. Triangle congruence by Side-Side-Side (SSS)
7. VUTS is a parallelogram 4. A quadrilateral with both pairs of opposite sides congruent and parallel is a parallelogram (definition of a parallelogram)
Therefore, we have proven that VUTS is a parallelogram.
To prove that quadrilateral VUTS is a parallelogram, we can use the given information that triangle SVX is congruent to triangle UTX and line segment SV is parallel to line segment TU. Here's how you can complete the two-column proof:
Statement 1: Triangle SVX is congruent to triangle UTX
Reason 1: Given
Statement 2: Line segment SV is parallel to line segment TU
Reason 2: Given
To prove that VUTS is a parallelogram, we need to show that opposite sides are parallel.
Statement 3: Line segment SV is parallel to line segment TU
Reason 3: Given (same as reason 2)
Since line segment SV is parallel to line segment TU, and triangle SVX is congruent to triangle UTX, we can conclude that line segment VT is parallel to line segment US. This is because corresponding sides of congruent triangles are parallel.
Statement 4: VUTS is a parallelogram
Reason 4: Opposite sides are parallel (line segment VT is parallel to line segment US)
To complete the proof, you need to use the properties of congruent triangles and the fact that parallel lines have corresponding sides that are parallel. Make sure to provide a clear statement and reason for each step to demonstrate your understanding of the concept.