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: Recall isosceles trapezoid properties

An isosceles trapezoid has one - pair of parallel sides. The non - parallel sides are congruent, and the diagonals are congruent. Also, base - angles are congruent.

Step2: Analyze each option

• $\overline{BC}\parallel\overline{AD}$: By definition, a trapezoid has one pair of parallel sides. In an isosceles trapezoid $ABCD$, $\overline{BC}\parallel\overline{AD}$ is correct.
• $\overline{BD}\perp\overline{AC}$: There is no property that says the diagonals of an isosceles trapezoid are perpendicular. This is not always true.
• $\overline{BA}\cong\overline{CD}$: In an isosceles trapezoid, the non - parallel sides are congruent. So $\overline{BA}\cong\overline{CD}$ is correct.
• $\overline{BE}\cong\overline{ED}$: Since the diagonals of an isosceles trapezoid are congruent ($AC = BD$) and the trapezoid is symmetric about the line of symmetry passing through the mid - points of the bases, $\overline{BE}\cong\overline{ED}$ is correct.
• $\angle CBA\cong\angle BCD$: In an isosceles trapezoid, base - angles are congruent. But $\angle CBA$ and $\angle BCD$ are not base - angles. This is not always true.

Answer:

$\overline{BC}\parallel\overline{AD}$, $\overline{BA}\cong\overline{CD}$, $\overline{BE}\cong\overline{ED}$