Question

Solve the following problem. Show your work to receive full credit for the problem. (1.5 points) In kite WXYZ, m∠WXY = 104°, and m∠VYZ = 49°. Find each measure. a. m<VZY = b. m<VXW = c. m<VWZ

Answer 1

To solve the problem involving kite WXYZ, we will use properties of kites and the measures of angles given.

1. **Understanding the Properties of Kites**:
- In a kite, two pairs of adjacent sides are equal. For kite WXYZ, sides WX = XY and WZ = YZ.
- The angles opposite the equal sides are equal. Hence, ∠WXY and ∠WZY are opposite angles with respect to the equal sides WZ and YZ.
- The diagonals of a kite intersect at right angles.

2. **Given Angles**:
- m∠WXY = 104°
- m∠VYZ = 49° (assuming this was a typo, and you meant ∠XYZ, as V isn't identified in the kite)

3. **Finding m∠WYZ**:
- Since the angles in a triangle sum up to 180°, we can find ∠XYZ:
- In triangle XYZ, we know that:
\[
m∠XYZ + m∠YZW + m∠WYZ = 180°
\]
- Since ∠WXY and ∠YZW are adjacent, the angles at vertex Y (∠XYZ and ∠WYZ) can be determined:
- m∠WYZ = m∠WXY (as the opposite angles should be equal in kites):
- Therefore,
\[
m∠WYZ = m∠YZW = 49° \quad (because \ m∠YZW + m∠WZY = 180° - 104°)
\]
- Thus, we have:
\[
m∠WYZ = 180° - 49° - 104° = 27°
\]

4. **Finding Each Angle**:
- a. To find **m∠VZY** (which is the same as ∠WZY):
\[
m∠WZY = 180° - (m∠WXY + m∠WYZ) \\
m∠WZY = 180° - (104° + 27°) \\
= 49°
\]

- b. To find **m∠VXW** (which is the same as ∠WXY):
\[
m∠WXZ = m∠WXY = 104°
\]

- c. To find **m∠VWZ** (which is the same as ∠WZY):
\[
m∠VWZ = 27° \\
(from what we calculated in point 3, ∠WZY = 27°)
\]

5. **Final Answers**:
- a. \(m∠VZY = 49°\)
- b. \(m∠VXW = 104°\)
- c. \(m∠VWZ = 27°\)

This shows all the necessary work and conclusions based on the properties of the kite!