Question

parallelogram abcd is a rectangle.
which statements are true?
select each correct answer.
2ax = bd
bc = cx
\\(\overline{ad} \perp \overline{cd}\\)
\\(m\angle abx = m\angle cbx\\)

Explanation:

Step1: Analyze \(2AX = BD\)

In a rectangle, diagonals are equal and bisect each other. So \(AC = BD\) and \(AX=\frac{1}{2}AC\) (since \(X\) is the midpoint of diagonals). Thus \(AX=\frac{1}{2}BD\), so \(2AX = BD\). This statement is true.

Step2: Analyze \(BC = CX\)

\(CX\) is half of the diagonal \(AC\) (as diagonals bisect each other), and \(BC\) is a side of the rectangle. There's no reason for \(BC\) and \(CX\) to be equal. This statement is false.

Step3: Analyze \(\overrightarrow{AD}\perp\overrightarrow{CD}\)

In a rectangle, all angles are right angles. So \(\angle ADC = 90^\circ\), which means \(\overrightarrow{AD}\perp\overrightarrow{CD}\). This statement is true.

Step4: Analyze \(m\angle ABX=m\angle CBX\)

\(BX\) is a diagonal bisector, but in a rectangle, \(AB
eq BC\) (unless it's a square), so the angles \(\angle ABX\) and \(\angle CBX\) are not equal. This statement is false.

Answer:

\(2AX = BD\), \(\overrightarrow{AD}\perp\overrightarrow{CD}\) (i.e., the first and the third options are correct)