Question

1. given: abcd is a rectangle

prove: the diagonals are congruent
(\\( \overline { b d } \cong \overline { a c } \\))

Explanation:

Step1: Properties of a rectangle

In rectangle \(ABCD\), \(AB = CD\) (opposite sides of a rectangle are equal), \(\angle ABC=\angle DCB = 90^{\circ}\), and \(BC=CB\) (common side).

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

In \(\triangle ABC\) and \(\triangle DCB\), we have:

• \(AB = CD\) (from the property of a rectangle)
• \(\angle ABC=\angle DCB\) (right angles)
• \(BC = CB\) (common side)

So, \(\triangle ABC\cong\triangle DCB\) (by SAS congruence).

Step3: Corresponding parts of congruent triangles

Since \(\triangle ABC\cong\triangle DCB\), then \(AC = BD\) (corresponding parts of congruent triangles are equal).

Answer:

Statement	Justification
\(\angle ABC=\angle DCB = 90^{\circ}\)	Definition of a rectangle (all angles are right angles)
\(BC = CB\)	Common side
\(\triangle ABC\cong\triangle DCB\)	SAS (Side - Angle - Side) congruence criterion
\(AC = BD\)	Corresponding parts of congruent triangles (\(\triangle ABC\) and \(\triangle DCB\)) are equal