Question

question 23 of 25 mary states, \if the diagonals of a parallelogram are congruent, then the parallelogram is a rectangle.\ decide if her statement is true or false. a. true b. false

Explanation:

Step1: Recall parallelogram properties

One of the properties of a rectangle is that its diagonals are congruent. Also, in a parallelogram, if the diagonals are congruent, we can prove it is a rectangle using triangle - congruence. Consider two adjacent triangles formed by the diagonals in a parallelogram. In a parallelogram \(ABCD\) with diagonals \(AC\) and \(BD\) such that \(AC = BD\), we can show that \(\triangle ABC\cong\triangle DCB\) (by SSS congruence criterion since \(AB = DC\) (opposite sides of parallelogram), \(BC=CB\) (common side) and \(AC = BD\)). Then \(\angle ABC=\angle DCB\). Since \(AB\parallel DC\) in parallelogram \(ABCD\), \(\angle ABC+\angle DCB = 180^{\circ}\). So \(\angle ABC=\angle DCB = 90^{\circ}\). A parallelogram with one right - angle is a rectangle.

Answer:

A. True