Given UVWX is a parallelogram and m∠WXV = 17°, m∠WVX = 29°, XW = 41, UX = 24 and UY = 15 find... a. m∠WVU = __ b. WV = __ c. m∠XUV = __ d. UW = __

Steps to Solve

1. Find the measure of angle ( \angle WUV )

In a parallelogram, opposite angles are equal. Therefore, since ( m\angle WXY = 29^\circ ), we have:

$$ m\angle WUV = 29^\circ $$

2. Calculate the measure of angle ( \angle XUV )

Using the property that the sum of angles around point ( U ) (in triangle ( XUV )) is ( 180^\circ ):

$$ m\angle XUV = 180^\circ - m\angle WUV - m\angle WXY $$

Substituting the known values:

$$ m\angle XUV = 180^\circ - 29^\circ - 17^\circ = 134^\circ $$

3. Find the length of side ( WV ) using the Law of Sines in triangle ( WUV )

Using the Law of Sines:

$$ \frac{WV}{\sin(17^\circ)} = \frac{UW}{\sin(29^\circ)} $$

Let ( UW = 15 ). Then,

$$ WV = \frac{15 \cdot \sin(17^\circ)}{\sin(29^\circ)} $$

4. Calculate the length of side ( UW )

Given that ( m\angle XUV ) has been calculated and we have triangles where the lengths are known, use the previously found measurements to determine ( UW ):

From the properties of triangles, we already know ( UW = 15 ) from the given diagram.

5. Calculate the length of side ( UW )

Utilizing the pythagorean theorem in triangle ( WYX ):

$$ UW = \sqrt{(WY)^2 + (XZ)^2} = 41 $$

So, knowing:

$$ UW = 15 $$