Question

which statements are correct regarding the properties of trapezoids? check all that apply. the diagonals of an isosceles trapezoid are congruent. the bases of a trapezoid are parallel. the adjacent sides of a trapezoid are congruent. base angles of a trapezoid are congruent. the diagonals of a trapezoid are perpendicular.

Explanation:

• The diagonals of an isosceles trapezoid are congruent: This is a well - known property of isosceles trapezoids. In an isosceles trapezoid \(ABCD\) with \(AD\parallel BC\) and \(AB = CD\), we can prove \(\triangle ABC\cong\triangle DCB\) (by \(SAS\) criterion since \(AB = CD\), \(\angle ABC=\angle DCB\), \(BC = CB\)), and then \(AC = BD\) (corresponding parts of congruent triangles).
• The bases of a trapezoid are parallel: By the definition of a trapezoid, a trapezoid is a quadrilateral with exactly one pair of parallel sides. These parallel sides are called the bases.
• The adjacent sides of a trapezoid are congruent: This is not a general property of trapezoids. In a non - isosceles trapezoid, the non - parallel sides (legs) are not congruent, and there is no requirement for the adjacent sides (one base and one leg) to be congruent.
• Base angles of a trapezoid are congruent: This is only true for isosceles trapezoids. In a non - isosceles trapezoid, the base angles are not congruent.
• The diagonals of a trapezoid are perpendicular: This is not a general property of trapezoids. There is no proof or general rule that the diagonals of a trapezoid (even an isosceles trapezoid, where diagonals are congruent but not necessarily perpendicular) are perpendicular.

Answer:

• The diagonals of an isosceles trapezoid are congruent.
• The bases of a trapezoid are parallel.