Question

use the image to answer the question. 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. (4 points)

Explanation:

Step1: State properties of a rectangle

A rectangle has four right angles and opposite sides are equal. So, \(LM = ON\), \(LO=MN\), and \(\angle LOM=\angle ONM = 90^{\circ}\) (Definition of a rectangle).

Step2: Apply the Side - Angle - Side (SAS) congruence criterion

In \(\triangle LOM\) and \(\triangle MNL\), we have \(LO = MN\) (opposite sides of rectangle), \(\angle LOM=\angle MNL = 90^{\circ}\), and \(OM = NL\) (opposite sides of rectangle). So, \(\triangle LOM\cong\triangle MNL\) (SAS congruence criterion).

Step3: Use congruent triangles to find congruent diagonals

Since \(\triangle LOM\cong\triangle MNL\), their corresponding parts are congruent. The diagonals \(LN\) and \(OM\) are corresponding parts of \(\triangle LOM\) and \(\triangle MNL\). So, \(LN\cong OM\) (Corresponding parts of congruent triangles are congruent).

Answer:

The diagonals \(LN\) and \(OM\) of rectangle \(LMNO\) are congruent.