To prove that the diagonals of rectangle LMNO are congruent using a two-column proof, we will consider the segments \( LP \), \( NP \), \( MP \), and \( OP \), where \( P \) is the intersection of diagonals \( LN \) and \( MO \).
Two-Column Proof:
| Statement | Reason | |---------------------------------------------------------------------------------|-----------------------------------------------------| | 1. Rectangle \( LMNO \) has vertices \( L, M, N, O \) arranged in order. | Definition of a rectangle. | | 2. Diagonal \( LN \) connects vertices \( L \) and \( N \). | By definition of a diagonal. | | 3. Diagonal \( MO \) connects vertices \( M \) and \( O \). | By definition of a diagonal. | | 4. \( P \) is the intersection of diagonals \( LN \) and \( MO \). | Given that \( P \) is the midpoint of the rectangle. | | 5. \( LP \) is a segment of diagonal \( LN \) and \( NP \) is another segment. | Definition of segments. | | 6. \( MP \) is a segment of diagonal \( MO \) and \( OP \) is another segment. | Definition of segments. | | 7. \( LP \cong NP \) because \( P \) is the midpoint of diagonal \( LN \). | Definition of a midpoint (midpoints divide segments into two congruent segments). | | 8. \( MP \cong OP \) because \( P \) is the midpoint of diagonal \( MO \). | Definition of a midpoint (midpoints divide segments into two congruent segments). | | 9. Triangles \( LPN \) and \( MOP \) are congruent by SSS (Side-Side-Side). | Since \( LP \cong OP \), \( NP \cong MP \), and \( LN \cong MO \) as opposite sides of rectangle. | | 10. Therefore, \( LN \cong MO \). | Corresponding parts of congruent triangles are congruent (CPCTC). |
Conclusion:
The diagonals \( LN \) and \( MO \) of rectangle \( LMNO \) are congruent, \( LN \cong MO \).
This proof demonstrates the congruence of the diagonals in a rectangle using geometric properties of congruent triangles.
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