We want to find the measure of the angles XPY, XPQ, and PXQ. Let's find them one at a time.
Angle XPY
The angle XPY is a central angle of the square. The measure of each central angle of a regular n-gon is 360n. For a square, we have that n=4. Let's substitute this value in the formula and simplify.
The segment PQ is an apothem in the square. The apothem bisects the vertex angle of the isosceles triangle formed by the radii. Since we know that the measure of this angle is 90^(∘), we can divide 90 by 2 to obtain the measure of angle XPQ.
90/2=45^(∘)
The measure of angle XPQ is 45^(∘).
Angle PXQ
As previously mentioned, the apothem bisects the vertex angle of the isosceles triangle formed by the radii. We can see in the diagram that the apothem makes a right angle with the side of the square. Therefore, we know the measures of two of the three interior angles of the right triangle XPQ. The missing angle is ∠ PXQ.
To find the measure of angle PXQ we will use the Triangle Sum Theorem. This theorem states that the sum of the three interior angles of a triangle adds to 180^(∘).
90+45+m∠PXQ=180^(∘)
⇕
m∠PXQ=45^(∘)
The measure of angle PXQ is 45^(∘).
