Prove that the diagonals of rectangle LMNO are congruent. You can write a two-column proof or a paragraph proof, but be sure to include a reason for each of your statements...

Answer 1

Here’s a paragraph proof to demonstrate that the diagonals of rectangle LMNO are congruent:

Proof:

In rectangle LMNO, we identify the points L, M, N, and O as the vertices. By definition, a rectangle is a quadrilateral with four right angles and opposite sides that are equal in length.

1. By definition, opposite sides of rectangle LMNO are equal, so \(LM = ON\) and \(LO = MN\).
2. The diagonals of rectangle LMNO are \(LN\) and \(MO\).
3. Since rectangle LMNO has four right angles, triangles \(LMO\) and \(LNO\) can be formed by drawing the diagonals \(LN\) and \(MO\).
4. In triangles \(LMO\) and \(LNO\), corresponding sides are equal: \(LM = ON\) and \(LO = MN\) (opposite sides of the rectangle).
5. Angle \(LMO\) is congruent to angle \(LNO\) since they are both right angles (each measuring 90 degrees).
6. By the Side-Angle-Side (SAS) congruence postulate (two sides and the included angle of one triangle are congruent to two sides and the included angle of another triangle), we can conclude that triangles \(LMO\) and \(LNO\) are congruent.
7. Consequently, corresponding parts of congruent triangles are congruent, which means \(LN = MO\).

Therefore, we have proven that the diagonals of rectangle LMNO, \(LN\) and \(MO\), are congruent.