Question

find the measure of the three missing angles in the parallelogram below.
image of a parallelogram with one angle labeled 105°, and the other three angles labeled x°, y°, z° respectively

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 \(x\) and \(y\) (consecutive to \(105^\circ\))

Since consecutive angles in a parallelogram are supplementary, \(x + 105^\circ= 180^\circ\) and \(y + 105^\circ= 180^\circ\).
Solving for \(x\): \(x = 180^\circ - 105^\circ = 75^\circ\).
Solving for \(y\): \(y = 180^\circ - 105^\circ = 75^\circ\).

Step3: Find angle \(z\) (opposite to \(105^\circ\))

In a parallelogram, opposite angles are equal. So \(z = 105^\circ\) (since \(z\) is opposite the \(105^\circ\) angle).

Answer:

\(x = 75^\circ\), \(y = 75^\circ\), \(z = 105^\circ\)