Given:modifying above S V with bar is parallel to modifying above T U with bar and Triangle Upper S Upper V Upper X is congruent to triangle Upper U Upper T Upper X
Prove: VUTS is a parallelogram.
Quadrilateral upper S upper T upper U upper V is shown with diagonals upper S upper U and upper T upper V intersecting at point upper X.
Write a paragraph proof.
First, since triangle SUX is congruent to triangle TUX, we know that angle SUX is congruent to angle TUX, angle UTX is congruent to angle USX, and side US is congruent to side UT (by CPCTC). Since the diagonals ST and UV intersect at X, we can conclude that angle SUX is congruent to angle TXV (by vertical angles).
Next, we are given that modifying angle S with a bar is parallel to modifying angle T with a bar. This means that angle SUX is congruent to angle UTV (by corresponding angles). Since triangle SUX is congruent to triangle TUX, we can also conclude that angle UTV is congruent to angle USX and angle UTX is congruent to angle TXV (by CPCTC).
Therefore, we have angle SUX is congruent to angle TXV, angle UTV is congruent to angle USX, and angle UTX is congruent to angle TXV. This shows that all opposite angles of quadrilateral VUTS are congruent, which is a property of a parallelogram.
Hence, by proving that all opposite angles of quadrilateral VUTS are congruent, we have shown that VUTS is a parallelogram.