Question

given: abcd is a square. prove: \\( \overline { ac } \perp \overline { bd } \\).

Explanation:

• In a square, all sides are congruent. So, \(AB\) (since \(ABCD\) is a square, \(AB = AD\)) is congruent to \(AD\).
• The reflexive property of congruence states that any segment is congruent to itself. So, \(AE\cong AE\) by the reflexive property.
• The diagonals of a square bisect each other. So, \(DE\cong BE\).
• The SSS (Side - Side - Side) congruence criterion states that if three sides of one triangle are congruent to three sides of another triangle, then the two triangles are congruent. Here, in \(\triangle AEB\) and \(\triangle AED\), \(AB = AD\), \(AE=AE\), \(BE = DE\), so \(\triangle AEB\cong\triangle AED\) by SSS.
• If two angles form a linear pair and are congruent, then each angle is \(90^{\circ}\) (definition of perpendicular lines: if the measure of the angle between two lines is \(90^{\circ}\), the lines are perpendicular).

Answer:

1. \(AB\)
2. reflexive property
3. \(BE\)
4. SSS
5. definition of perpendicular lines