Two-Column Proof
| Statement | Reason | |---------------------------------------------------|--------------------------------------| | 1. Let rectangle LMNO be defined by vertices L, M, N, O. | Definition of a rectangle | | 2. Diagonal LN and diagonal MO are drawn. | Construct the diagonals | | 3. In rectangle LMNO, opposite sides are equal: LM = ON and LN = MO. | Properties of rectangles | | 4. Additionally, side LO = side MN and side LM = side ON. | Properties of rectangles | | 5. Triangle LMO and triangle NMO are formed by the diagonals. | Formation of triangles using diagonals | | 6. In triangle LMO, LM = ON and LO = MN. | Given sides of rectangle | | 7. Diagonals LN and MO common to both triangles. | Common side in triangles | | 8. Thus, by the Side-Side-Side (SSS) congruence criterion, triangle LMO is congruent to triangle NMO. | SSS Congruence Theorem | | 9. Since the triangles are congruent, their corresponding parts are equal, so LN = MO. | Corresponding Parts of Congruent Triangles | | 10. Therefore, the diagonals of rectangle LMNO are congruent (LN ≅ MO). | Definition of congruence |
Paragraph Proof
In rectangle LMNO, we begin by noting that opposite sides are equal, specifically that \( LM = ON \) and \( LO = MN \). By drawing the diagonals \( LN \) and \( MO \), we can form two triangles, \( \triangle LMO \) and \( \triangle NMO \). For these triangles, we observe that \( LM \) is equal to \( ON \) and \( LO \) is equal to \( MN \). Furthermore, the sides \( LN \) and \( MO \) are common to both triangles. By the Side-Side-Side (SSS) congruence criterion, we can conclude that \( \triangle LMO \cong \triangle NMO \). Therefore, since corresponding parts of congruent triangles are equal, it follows that the diagonals \( LN \) and \( MO \) are congruent \( (LN \cong MO) \). Thus, we have proved that the diagonals of rectangle LMNO are congruent.
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