Question

if quadrilateral abcd is an isosceles trapezoid, which statements must be true? select three options

bc || ad
bd ⊥ ac
ba ≅ cd
be ≅ ed
∠cba ≅ ∠bcd

Explanation:

Step1: Properties of isosceles trapezoid

An isosceles trapezoid has one pair of parallel sides (bases). Here, \(BC\parallel AD\). Also, non - parallel sides (legs) are congruent, so \(BA\cong CD\).

Step2: Diagonals of isosceles trapezoid

In an isosceles trapezoid \(ABCD\) with \(BC\parallel AD\), diagonals \(AC\) and \(BD\) are congruent. Since \(\triangle BEC\cong\triangle DEA\) (by AAS, as \(BC\parallel AD\) gives \(\angle EBC=\angle EDA\) and \(\angle ECB=\angle EAD\) and \(BC = AD\) (from the property of isosceles trapezoid)), then \(BE\cong ED\).

Step3: Angles of isosceles trapezoid

Base angles of an isosceles trapezoid are congruent. For base \(BC\), \(\angle CBA\) and \(\angle BCD\) are base angles (when \(AB = CD\) and \(BC\) is a base), so \(\angle CBA\cong\angle BCD\).
The statement \(BD\perp AC\) is not a property of an isosceles trapezoid.

Answer:

\(\overline{BC}\parallel\overline{AD}\), \(\overline{BA}\cong\overline{CD}\), \(\angle CBA\cong\angle BCD\)