Question

∠acd ≅ ∠abd
ad ⊥ cb
select all statements that are true about equilateral triangle abc.
a angles b and c are 60 degrees.
b ( x = 3sqrt{3} )
c ( x = 6sqrt{3} )
d triangle abd is congruent to triangle acd.
e segments bd and cd are both 3 units long.

Explanation:

Step1: Analyze Option A

In an equilateral triangle, all angles are \(60^\circ\). So \(\angle B=\angle C = 60^\circ\), so A is true.

Step2: Analyze Option B and C

Since \(AD\perp CB\) and \(\triangle ABC\) is equilateral with side length \(6\), \(CD = BD=\frac{6}{2}=3\) (by property of equilateral triangle, altitude bisects the base). In right triangle \(ACD\), using Pythagorean theorem: \(x=\sqrt{AC^{2}-CD^{2}}=\sqrt{6^{2}-3^{2}}=\sqrt{36 - 9}=\sqrt{27}=3\sqrt{3}\), so B is true, C is false.

Step3: Analyze Option D

In \(\triangle ABD\) and \(\triangle ACD\), \(AD = AD\) (common side), \(AB=AC\) (sides of equilateral triangle), \(BD = CD\) (proven above), so by SSS congruence, \(\triangle ABD\cong\triangle ACD\), so D is true.

Step4: Analyze Option E

Since \(AD\) is altitude of equilateral triangle with side \(6\), it bisects \(CB\), so \(BD=CD=\frac{6}{2}=3\), so E is true.

Answer:

A. Angles \(B\) and \(C\) are \(60\) degrees.
B. \(x = 3\sqrt{3}\)
D. Triangle \(ABD\) is congruent to triangle \(ACD\).
E. Segments \(BD\) and \(CD\) are both \(3\) units long.