Question

question 2 of 10

given that \\(abcd\\) is a rhombus, what is the value of \\(x\\)?

\\(b\\) \\((4x - 25)^\circ\\) \\(c\\)
\\(x^\circ\\) \\(a\\) \\(d\\)

a. 29
b. 23
c. 46
d. 38
e. 45
f. cannot be determined

Explanation:

Identify properties of a rhombus

A rhombus is a parallelogram with four congruent sides. In any rhombus, the diagonals bisect the opposite angles. This means that diagonal \(AC\) bisects angle \(\angle BAD\) and angle \(\angle BCD\), and diagonal \(BD\) bisects angle \(\angle ABC\) and angle \(\angle ADC\).

Relate the given angles

Since \(ABCD\) is a rhombus, the opposite sides \(BC\) and \(AD\) are parallel (\(BC \parallel AD\)).
The diagonal \(AC\) acts as a transversal intersecting these parallel lines.
By the alternate interior angles theorem, we have:

$$\angle BCA \cong \angle DAC$$

From the given diagram:

• The measure of \(\angle BCA\) is \((4x - 25)^\circ\).
• The measure of \(\angle DAC\) is \(x^\circ\).

Set up the equation

Equating the measures of the alternate interior angles:

$$4x - 25 = x$$

Solve for x

Subtract \(x\) from both sides:

$$3x - 25 = 0$$

Add \(25\) to both sides:

$$3x = 25$$

Divide by \(3\):

$$x = \frac{25}{3} \approx 8.33$$

Re-evaluate the angle labels

Let's re-examine the diagram carefully.
The angle labeled \((4x - 25)^\circ\) is \(\angle DBC\).
The angle labeled \(x^\circ\) is \(\angle DAC\).
Let \(O\) be the intersection of the diagonals \(AC\) and \(BD\).
In a rhombus, the diagonals are perpendicular to each other, so \(\angle BOC = \angle AOD = 90^\circ\).
In right triangle \(\triangle BOC\), the acute angles are \(\angle OBC = (4x - 25)^\circ\) and \(\angle BCO\).
Since \(BC \parallel AD\), the alternate interior angle to \(\angle DAC\) (which is \(\angle OAD = x^\circ\)) is \(\angle OCB = x^\circ\).
Therefore, in the right-angled triangle \(\triangle BOC\) at \(O\):

$$\angle OBC + \angle OCB = 90^\circ$$

$$(4x - 25) + x = 90$$

Solve the correct equation

Combine like terms:

$$5x - 25 = 90$$

Add \(25\) to both sides:

$$5x = 115$$

Divide by \(5\):

$$x = 23$$

Answer:

• (A) 29
• (B) 23 (Correct answer)
• (C) 46
• (D) 38
• (E) 45
• (F) Cannot be determined