Let's break down the questions you've provided and solve them step-by-step:

(11) The center of the circle $O$ is located on the points $A$, $B$, $C$, and $D$ on the circle. $BD$ is perpendicular to the diameter $AC$.

i. Take $\angle A\hat{C}B = x$ and find the value of the following angles in terms of $x$, with reasons:

We are given that $BD$ is perpendicular to the diameter $AC$, and we need to express various angles in terms of $x$. Let's begin with each angle step-by-step:

1. Angle $AOB$:

   • $O$ is the center of the circle, meaning $AO$ and $BO$ are radii of the circle.
   • Therefore, $\triangle AOB$ is isosceles, and the angle $AOB$ subtends the arc $AB$ at the center.
   • By the Central Angle Theorem, the angle subtended by an arc at the center of the circle is twice the angle subtended at the circumference.
   • Hence, $\angle AOB = 2 \times \angle A\hat{C}B = 2x$.

2. Prove that $\triangle ABC$ is an isosceles triangle:

   • In a circle, the angles subtended by the same arc are equal.
   • The angle $\angle ACB = x$ is subtended by arc $AB$.
   • Similarly, the angle $\angle ABC$ also subtends arc $AB$, so $\angle ABC = x$.
   • Since two angles in $\triangle ABC$ are equal, the triangle is isosceles.
   • Hence, $\triangle ABC$ is isosceles with $AB = AC$.

3. Show that $\angle ADB = \angle BDC$:

   • Both angles $\angle ADB$ and $\angle BDC$ subtend the same arc $BD$.
   • By the Inscribed Angle Theorem, angles subtended by the same arc are equal.
   • Therefore, $\angle ADB = \angle BDC$.

Part b and c: $\angle ADB$ and $\angle BAD$

4. Find $\angle ADB$:

   • As we know from the above, $\angle ADB$ is equal to $\angle BDC$ because they subtend the same arc.
   • To determine $\angle ADB$, we would need more specific information about the diagram. However, based on the context of the circle and perpendicular diameters, it's clear that this angle can be directly related to the geometry of the circle.

5. Find $\angle BAD$:

   • To find $\angle BAD$, we need to analyze its relationship with the geometry of the circle. Depending on the diagram, this could be calculated using the known values of the other angles, especially since $BD$ is perpendicular to the diameter.

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Would you like further clarification or a more detailed diagram explanation?

Here are 5 questions to help you practice:

1. How does the Central Angle Theorem help find $\angle AOB$?
2. Why is $\triangle ABC$ isosceles based on angle properties?
3. How does the Inscribed Angle Theorem prove that $\angle ADB = \angle BDC$?
4. How does perpendicularity affect the relationships between angles in this setup?
5. How would the diagram change if $BD$ were not perpendicular to $AC$?

Tip: Always remember that the central angle is twice any inscribed angle that subtends the same arc. This is key for solving many circle theorems.