Let's solve the given problem step by step:

(i) Explain why ∠APC = 90°.

The quadrilateral $ABCD$ is a square, and it is inscribed in a circle. A square inscribed in a circle implies that its diagonals are diameters of the circle. The diagonal $AC$ is the diameter of the circle.

By the angle in a semicircle theorem, any angle subtended by a diameter (like $\angle APC$) on the circumference of the circle is $90^\circ$. Hence:

$\boxed{\angle APC = 90^\circ.}$

(ii) Explain why ∠APB = ∠BPC.

The square $ABCD$ is cyclic, meaning all its vertices lie on the same circle.

• $P$ is a point on the circumference, and $A$, $B$, and $C$ are also points on the circle.
• $\angle APB$ and $\angle BPC$ subtend the same arc, $AB$, of the circle.
• In a cyclic quadrilateral, angles subtending the same chord or arc are equal.

Thus, $\boxed{\angle APB = \angle BPC}$.

(iii) (a) Find ∠APB.

Since $ABCD$ is a square, each of its interior angles is $90^\circ$. The diagonals divide these angles into two equal parts, each $45^\circ$. Therefore, $\angle APB = 45^\circ$.

$\boxed{\angle APB = 45^\circ.}$

(iii) (b) Find ∠APD.

• The angle $\angle APD$ subtends the arc $AD$, which is $90^\circ$ (one-fourth of the circle).
• In a circle, the angle subtended by an arc at the circumference is half the central angle subtended by the same arc.
• Therefore, $\angle APD = \frac{1}{2} \times 90^\circ = 45^\circ$.

$\boxed{\angle APD = 45^\circ.}$

(iv) PC and AD intersect at R. Given that ∠ARC = 127°, find ∠PDC.

To solve this:

1. $\angle ARC = 127^\circ$. This angle subtends the arc $PC$ of the circle.
2. The angle $\angle PDC$ is subtended by the same arc $PC$ but on the opposite side of the circle.
3. By the cyclic quadrilateral property, opposite angles subtending the same arc sum to $180^\circ$.

Thus: $\angle PDC + \angle ARC = 180^\circ.$ $\angle PDC = 180^\circ - 127^\circ = 53^\circ.$

$\boxed{\angle PDC = 53^\circ.}$

Recap of Answers:

1. $\angle APC = 90^\circ.$
2. $\angle APB = \angle BPC.$
3. $\angle APB = 45^\circ.$
4. $\angle APD = 45^\circ.$
5. $\angle PDC = 53^\circ.$

Let me know if you want detailed steps for any part!
Relative Questions:

1. Why is the angle in a semicircle always $90^\circ$?
2. What are the properties of a cyclic quadrilateral?
3. How do the angles in a square relate to the circle it is inscribed in?
4. Can we generalize the concept of angles subtending arcs in other polygons?
5. How does the cyclic property apply to other intersecting chords?

Tip: Always look for key properties like symmetry, semicircles, and arcs in geometry problems involving circles!