find the values of x, y, and z in the parallelogram. note that the figure is not drawn to scale.

Explanation:

Step1: Identify alternate interior angles

In a parallelogram, opposite sides are parallel, so alternate interior angles are equal. $$y = 29$$

Step2: Use alternate interior angles for x

Similarly, the angle labeled $56^{\circ}$ and the angle $x$ are alternate interior angles. $$x = 56$$

Step3: Calculate the third angle in the central triangle

The angle supplementary to $102^{\circ}$ along the diagonal is $180^{\circ} - 102^{\circ} = 78^{\circ}$. In the bottom triangle formed by the diagonals and the bottom side, the angles are $29^{\circ}$, $x^{\circ}$, and the angle at the intersection.

Step4: Use the sum of angles in a triangle

In the triangle containing $z$, the angle opposite $102^{\circ}$ is also $102^{\circ}$ (vertical angles). The angle alternate to $56^{\circ}$ is $x=56^{\circ}$. $$z + 102 + (\text{angle alternate to top-right}) = 180$$ Alternatively, use the large triangle on the left: $56 + 29 + y + (\text{angle at intersection}) = 180$. Using the triangle with $x$ and $z$: the angle adjacent to $102^{\circ}$ is $180^{\circ} - 102^{\circ} = 78^{\circ}$. In that triangle: $$x + z + 78 = 180$$

Step5: Solve for z

Substitute $x = 56$ into the equation. $$56 + z + 78 = 180$$ $$z + 134 = 180$$ $$z = 46$$

Answer:

x = 56, y = 29, z = 46