Question

what is the measure of angle bcd?

\\(25^\circ\\)
\\(40^\circ\\)
\\(140^\circ\\)
\\(155^\circ\\)

Explanation:

Construct parallel auxiliary lines

Let the horizontal line passing through \(A\) and \(B\) be \(L_1\), and the horizontal line passing through \(D\) and \(C\) be \(L_2\). Since both lines have parallel arrows, \(L_1 \parallel L_2\).
Construct a horizontal line \(L_3\) passing through \(A\) such that \(L_3 \parallel L_1 \parallel L_2\).
Construct a horizontal line \(L_4\) passing through \(B\) such that \(L_4 \parallel L_1 \parallel L_2\).

Determine the interior angles of the polygon

The line segment \(AD\) connects \(L_1\) at \(A\) and \(L_2\) at \(D\).
The angle at \(A\) inside the quadrilateral \(ABCD\) is composed of the given \(25^\circ\) angle and the angle between \(AB\) and the horizontal line \(L_1\).
The interior angle \(\angle DAB = 180^\circ - 25^\circ = 155^\circ\).
The interior angle \(\angle ADC = 180^\circ - 149^\circ = 31^\circ\).
The interior angle \(\angle ABC = 180^\circ - 146^\circ = 34^\circ\).

Calculate the measure of angle BCD

Using the polygon interior angle sum for a quadrilateral \(ABCD\):

$$ \angle DAB + \angle ADC + \angle ABC + \angle BCD = 360^\circ $$

$$ 155^\circ + 31^\circ + 34^\circ + \angle BCD = 360^\circ $$

$$ 220^\circ + \angle BCD = 360^\circ $$

$$ \angle BCD = 140^\circ $$

Answer:

• 25°
• 40°
• 140° (Correct answer)
• 155°