Question

find the measure of the three missing angles in the parallelogram below.
image of a parallelogram with one angle labeled 76°, and the other three angles labeled x°, y°, z°

Explanation:

Step1: Recall properties of parallelograms

In a parallelogram, opposite angles are equal, and consecutive angles are supplementary (sum to \(180^\circ\)).

Step2: Find angle \(x\)

Angle \(x\) and the \(76^\circ\) angle are consecutive angles? Wait, no, wait. Wait, in the parallelogram, let's label the vertices. Let's say the angles are at vertices: let's assume the given \(76^\circ\) is at the top, \(x\) at the left, \(y\) at the bottom, \(z\) at the right. Then, opposite angles: \(x\) and \(z\) are opposite? Wait, no, in a parallelogram, opposite angles are equal. Also, consecutive angles are supplementary. So, the angle adjacent to \(76^\circ\) (consecutive) would be \(180 - 76 = 104^\circ\). Wait, let's correct: in a parallelogram, adjacent angles (consecutive) are supplementary. So, if one angle is \(76^\circ\), then the angle adjacent to it (consecutive) is \(180 - 76 = 104^\circ\). Then, opposite angles are equal. So, let's see:

Looking at the diagram, the angle labeled \(76^\circ\) and angle \(y\) are opposite? Wait, no, maybe the diagram has the angles: let's see, the left angle is \(x\), top is \(76^\circ\), bottom is \(y\), right is \(z\). Then, in a parallelogram, opposite angles are equal: so \(x = z\), and \(76^\circ = y\)? Wait, no, that can't be. Wait, no, consecutive angles: so top angle \(76^\circ\), left angle \(x\) are consecutive, so \(76 + x = 180\), so \(x = 180 - 76 = 104^\circ\). Then, opposite angles: \(x = z = 104^\circ\), and \(76^\circ = y\), because opposite angles in parallelogram are equal. Wait, let's confirm:

Property 1: Opposite angles of a parallelogram are equal.

Property 2: Consecutive angles of a parallelogram are supplementary (sum to \(180^\circ\)).

So, let's identify the angles:

• Let the angle given be \(76^\circ\). Let's say it's angle \(A\). Then, angle \(C\) (opposite to \(A\)) is also \(76^\circ\). Angles \(B\) and \(D\) (opposite to each other) are equal, and each is \(180 - 76 = 104^\circ\), since consecutive angles are supplementary.

Looking at the diagram, the angles are labeled as \(x\), \(76^\circ\), \(y\), \(z\). So, let's assume:

• \(76^\circ\) and \(y\) are opposite angles (so \(y = 76^\circ\)).

• \(x\) and \(z\) are opposite angles (so \(x = z\)).

• \(76^\circ\) and \(x\) are consecutive angles (so \(76 + x = 180\)), so \(x = 180 - 76 = 104^\circ\). Then \(z = x = 104^\circ\), and \(y = 76^\circ\).

So:

Step1: Find \(x\) (consecutive to \(76^\circ\))

Since consecutive angles in a parallelogram are supplementary:
\(x + 76^\circ = 180^\circ\)
\(x = 180^\circ - 76^\circ = 104^\circ\)

Step2: Find \(y\) (opposite to \(76^\circ\))

Opposite angles in a parallelogram are equal:
\(y = 76^\circ\)

Step3: Find \(z\) (opposite to \(x\))

Opposite angles in a parallelogram are equal:
\(z = x = 104^\circ\)

Answer:

The missing angles are \(x = 104^\circ\), \(y = 76^\circ\), \(z = 104^\circ\)