Question

practice solving problems involving trapezoids and kites. 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:

• For "The diagonals of an isosceles trapezoid are congruent":
• By the definition and properties of an isosceles trapezoid, in an isosceles trapezoid \(ABCD\) with \(AD\parallel BC\) and \(AB = CD\), we can prove \(\triangle ABC\cong\triangle DCB\) (using \(AB = CD\), \(BC=CB\), \(\angle ABC=\angle DCB\) - base angles of isosceles trapezoid). Then \(AC = BD\) (corresponding parts of congruent triangles), so the diagonals of an isosceles trapezoid are congruent.
• For "The bases of a trapezoid are parallel":
• The definition of a trapezoid is a quadrilateral with exactly one pair of parallel sides. These parallel sides are called the bases of the trapezoid.
• For "The adjacent sides of a trapezoid are congruent":
• In a general trapezoid, there is no such property that adjacent sides are congruent. For example, in a non - isosceles trapezoid \(ABCD\) with \(AD\parallel BC\), \(AB

eq AD\) and \(CD
eq BC\) in most cases.

• For "Base angles of a trapezoid are congruent":
• This is only true for an isosceles trapezoid. In a non - isosceles trapezoid, the base angles are not congruent. For example, in a trapezoid \(ABCD\) with \(AD\parallel BC\) and \(AB

eq CD\), \(\angle ABC
eq\angle BAD\)

• For "The diagonals of a trapezoid are perpendicular":
• In a general trapezoid, there is no such property. In an isosceles trapezoid, the diagonals are congruent but not necessarily perpendicular. We can construct a counter - example of a non - right trapezoid (neither a right trapezoid nor a special trapezoid like a square or rhombus which are not trapezoids in the strict non - inclusive definition) where diagonals are not perpendicular.

Answer:

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