Question

question 20 of 27
what is the value of x for the parallelogram shown?
diagram of parallelogram with vertices b, c, d, a; angle at b is 4x°, angle at c is 60°
options:
a. 75
b. 30
c. 15
d. 60

Explanation:

Step1: Recall parallelogram angle property

In a parallelogram, consecutive angles are supplementary (sum to \(180^\circ\)). So, \(\angle B + \angle C = 180^\circ\).

Step2: Substitute angle values

We know \(\angle B = 4x^\circ\) and \(\angle C = 60^\circ\). So, \(4x + 60 = 180\).

Step3: Solve for x

Subtract 60 from both sides: \(4x = 180 - 60 = 120\). Then divide by 4: \(x=\frac{120}{4}=30\). Wait, no, wait—wait, in a parallelogram, adjacent angles are supplementary, but also, opposite angles are equal. Wait, maybe I made a mistake. Wait, in the parallelogram, \(\angle B\) and \(\angle C\) are adjacent? Wait, no, in the diagram, ABCD is a parallelogram, so AB is parallel to CD, and BC is parallel to AD. So angle B and angle C: are they consecutive? Wait, no, angle B and angle A are consecutive, angle B and angle C: wait, in a parallelogram, consecutive angles (adjacent) are supplementary. Wait, maybe the diagram: let's see, vertices are A, B, C, D in order. So AB is connected to B and A, BC to B and C, CD to C and D, DA to D and A. So angle at B is \(4x\), angle at C is \(60^\circ\). So AB is parallel to CD, so angle B and angle C: are they same - side interior angles? Wait, no, AB and CD are parallel, and BC is a transversal. So angle B and angle C should be supplementary? Wait, no, if AB || CD, then angle B + angle C = 180? Wait, no, angle B and angle A are supplementary, angle A and angle D are supplementary, angle D and angle C are supplementary, angle C and angle B are supplementary? Wait, no, in a parallelogram, opposite angles are equal, and consecutive angles are supplementary. So angle B and angle D are equal, angle A and angle C are equal? Wait, no, wait the standard property: In a parallelogram, opposite angles are equal, and consecutive angles (adjacent) are supplementary. So if angle C is \(60^\circ\), then angle A is also \(60^\circ\) (opposite angles). Then angle B and angle D are equal, and angle B + angle A = 180 (consecutive angles). So angle B = \(180 - 60=120^\circ\). But angle B is \(4x^\circ\), so \(4x = 120\), then \(x = 30\)? Wait, but the options have 30 as option B, but wait, maybe I mixed up the angles. Wait, maybe angle B and angle C are supplementary? Wait, let's re - examine the diagram. If ABCD is a parallelogram, then AB || CD, and BC is a side. So angle at B (∠B) and angle at C (∠C): are they adjacent? Yes, because B is connected to C. So in a parallelogram, adjacent angles are supplementary. So ∠B + ∠C = 180°. So \(4x+60 = 180\). Then \(4x=180 - 60 = 120\), so \(x = 30\). Wait, but let's check the options. Option B is 30. But wait, maybe I made a mistake. Wait, another property: in a parallelogram, opposite angles are equal. So if angle C is \(60^\circ\), then angle A is \(60^\circ\), and angle B and angle D are equal. Also, consecutive angles are supplementary, so angle B + angle A=180, so angle B = 120, which is \(4x\), so \(x = 30\). So the answer should be 30? Wait, but let's check the options again. Option B is 30. Wait, but maybe the diagram is different. Wait, maybe angle B and angle C are supplementary? Wait, no, in a parallelogram, adjacent angles are supplementary. So if the parallelogram is labeled A - B - C - D, then AB is parallel to CD, and BC is parallel to AD. So angle at B (∠B) and angle at C (∠C): AB || CD, BC is a transversal, so ∠B and ∠C are same - side interior angles, so they should be supplementary. So ∠B+∠C = 180. So \(4x + 60=180\), \(4x = 120\), \(x = 30\). So the answer is 30. Wait, but let's check the options. Option B is 30. So that's correct.

Wait, but wait, maybe I messed up the angle labels. Let's think again. In a parallelogram, opposite angles are equal. So if angle C is \(60^\circ\), then angle A is \(60^\circ\), and angle B and angle D are equal. Also, the sum of all interior angles in a quadrilateral is \(360^\circ\). So \(2\times(60 + 4x)=360\). Divide both sides by 2: \(60 + 4x = 180\). Then \(4x=120\), \(x = 30\). Yes, that's correct. So the value of x is 30.

Answer:

B. 30