Question

question
find the measure of the three missing angles in the parallelogram below.
answer
attempt 1 out of 2
$x = square$ $y = square$ $z = square$

Explanation:

Step1: Recall properties of parallelograms

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

Step2: Find \(x\)

\(x\) and \(113^\circ\) are consecutive angles? No, wait, \(x\) and \(113^\circ\)? Wait, no, in the parallelogram, opposite angles are equal, and consecutive angles are supplementary. Wait, the angle given is \(113^\circ\), let's see the diagram. The angle \(113^\circ\) and \(z\)? Wait, no, let's label: in a parallelogram, opposite angles are equal. So the angle adjacent to \(113^\circ\) (let's say \(z\)): consecutive angles sum to \(180^\circ\), so \(z + 113^\circ= 180^\circ\), so \(z = 180 - 113 = 67^\circ\)? Wait, no, wait, maybe I mixed up. Wait, in a parallelogram, opposite angles are equal. So the angle \(x\): is \(x\) opposite to \(z\)? Wait, no, let's look at the diagram. The given angle is \(113^\circ\), and the angle \(x\) is opposite to... Wait, maybe the angle \(113^\circ\) and \(y\) are opposite? No, wait, let's re-express.

Wait, in a parallelogram, opposite angles are equal. So if one angle is \(113^\circ\), its opposite angle is also \(113^\circ\). Wait, no, maybe the angle \(113^\circ\) and \(y\) are opposite? Wait, the diagram: the bottom angle is \(113^\circ\), the top right is \(x\), top left is \(y\), bottom left is \(z\). So in a parallelogram, opposite angles are equal. So bottom angle \(113^\circ\) and top angle \(y\) are opposite? No, wait, adjacent angles: consecutive angles are supplementary. So bottom angle \(113^\circ\) and bottom left angle \(z\): no, bottom angle is \(113^\circ\) (let's say the bottom right angle), so bottom left angle \(z\) and bottom right angle \(113^\circ\) are consecutive, so \(z + 113 = 180\), so \(z = 67\). Then \(x\) is opposite to \(z\)? No, \(x\) is top right, \(z\) is bottom left: opposite angles, so \(x = z = 67\)? Wait, no, that can't be. Wait, no, maybe the top right angle \(x\) and bottom right angle \(113^\circ\) are consecutive? Wait, no, sides are parallel. So in a parallelogram, \(AB \parallel CD\) and \(AD \parallel BC\). So angle at \(B\) (bottom right) is \(113^\circ\), angle at \(A\) (top right) is \(x\), angle at \(D\) (top left) is \(y\), angle at \(C\) (bottom left) is \(z\). Then angle at \(B\) and angle at \(A\) are consecutive, so \(x + 113 = 180\), so \(x = 67\). Then angle at \(A\) (\(x = 67\)) and angle at \(C\) (\(z\)) are opposite, so \(z = 67\). Angle at \(B\) (\(113\)) and angle at \(D\) (\(y\)) are opposite, so \(y = 113\). Ah, that makes sense. So:

Step1: Find \(x\) (consecutive to \(113^\circ\))

Consecutive angles in parallelogram sum to \(180^\circ\), so \(x + 113^\circ = 180^\circ\)
\(x = 180 - 113 = 67^\circ\)

Step2: Find \(y\) (opposite to \(113^\circ\))

Opposite angles in parallelogram are equal, so \(y = 113^\circ\)

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

Opposite angles in parallelogram are equal, so \(z = x = 67^\circ\)

Wait, let's verify: consecutive angles: \(x = 67\) and \(y = 113\): \(67 + 113 = 180\), correct. \(z = 67\) and \(113\): \(67 + 113 = 180\), correct. Opposite angles: \(x = z = 67\), \(y = 113\) (opposite to bottom right \(113\)), correct.

So:

\(x = 67^\circ\)

\(y = 113^\circ\)

\(z = 67^\circ\)

Wait, but let's check again. If the bottom right angle is \(113^\circ\), then the top left angle \(y\) is opposite to it, so \(y = 113^\circ\). The top right angle \(x\) is consecutive to \(y\), so \(x + y = 180\), so \(x = 180 - 113 = 67^\circ\). Then the bottom left angle \(z\) is opposite to \(x\), so \(z = x = 67^\circ\). Yes, that's corre…

Answer:

\(x = 67\), \(y = 113\), \(z = 67\)