To find the measures of the numbered angles in the kite, let's go over a few properties of kites and use the information provided:

Kite Properties:

1. A kite has two pairs of adjacent sides that are equal.
2. The diagonals of a kite are perpendicular to each other.
3. One pair of opposite angles (formed by the unequal sides) is equal, and the angles between the unequal sides are bisected by the diagonal.

Analyzing the Given Information:

• We are given that $\angle 5 = 52^\circ$.
• The diagonals intersect perpendicularly, so all four angles around the intersection point are right angles.

Steps to Find the Measures of Each Numbered Angle:

1. Angle 1 and Angle 3: Since the diagonals are perpendicular, the angles around the intersection point (angles 1 and 3) are $90^\circ$.

   $\angle 1 = 90^\circ, \quad \angle 3 = 90^\circ$

2. Angle 2 and Angle 4: These are the other two angles of the kite. In a kite, the opposite angles formed by the unequal sides (angles 2 and 4 in this case) are equal. Since we know that $\angle 5 = 52^\circ$, which forms part of one of these angles, the remaining part of each angle (angle 2 and angle 4) is also $52^\circ$.

   $\angle 2 = 52^\circ, \quad \angle 4 = 52^\circ$

Final Answer:

• $\angle 1 = 90^\circ$
• $\angle 2 = 52^\circ$
• $\angle 3 = 90^\circ$
• $\angle 4 = 52^\circ$
• $\angle 5 = 52^\circ$ (given)

Would you like further explanation on any of these steps, or do you have other questions?

Here are five related questions to consider:

1. What are the properties of the diagonals in different types of quadrilaterals?
2. How do you determine the measures of angles in other quadrilaterals like rhombuses or parallelograms?
3. How do the properties of a kite differ from those of a parallelogram?
4. How can the knowledge of symmetry help in solving angle problems in kites?
5. How would this problem change if it were a rhombus instead of a kite?

Tip: Remember that in a kite, the diagonals not only intersect at right angles but also one diagonal bisects the other, which can help solve for unknown lengths and angles.