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I see you've uploaded a set of geometry problems asking to calculate the value of "x" in each diagram. I will go through each problem and solve them systematically:

### Problem 1:
You are given a circle with a central angle of 35°. Since the angle at the center of the circle subtended by the same arc is twice the angle at the circumference, the angle subtended by the same arc at point \(x\) is half of 35°.
\[
x = \frac{35}{2} = 17.5^\circ
\]
However, none of the answers match 17.5°, so I'll assume there's a need for rounding or another method at play here. Let's double-check later on.

### Problem 2:
This is an inscribed angle theorem. The angle \(x\) is subtended by the same arc as the 70° angle outside the circle (external angle). Using the tangent-secant angle theorem:
\[
x = \frac{70}{2} = 35^\circ
\]
So, \(x = 35^\circ\).

### Problem 3:
You are asked to find \(x\) in a right triangle inscribed in a semicircle. According to the inscribed angle theorem, the angle subtended by the diameter of a circle is always 90° (right angle). Given a right triangle and one of the angles (32°), \(x\) is:
\[
x = 90^\circ - 32^\circ = 58^\circ
\]

### Problem 4:
The diagram shows two arcs, one of 80° and another of 32°. The angle \(x\) is inscribed and subtended by these arcs. The sum of the inscribed angles is equal to half the sum of the arcs:
\[
x = \frac{80^\circ + 32^\circ}{2} = \frac{112^\circ}{2} = 56^\circ
\]

### Problem 7:
You have a semicircle, and the angle of 32° is given. The angle at the triangle formed inside the semicircle is 90° because any angle subtended by a diameter is 90°:
\[
x = 90^\circ - 32^\circ = 58^\circ
\]
But the options don't include this value, so further confirmation or a different approach might be necessary.

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Would you like to continue with more detailed explanations for the rest, or should I clarify any of the above first? 

### Five Related Questions:
1. How do the inscribed angle theorem and the central angle theorem differ in a circle?
2. What are the properties of angles subtended by the same arc in a circle?
3. How does the tangent-secant theorem work in circles?
4. Can you explain how right angles in semicircles relate to the diameter?
5. What are the applications of the angle bisector theorem in triangle geometry?

### Tip:
In circle theorems, remember that angles subtended by the same arc at the circumference are equal, and angles subtended by a diameter are always right angles (90°).

Calculate the value of x in each of the geometry problems shown in the image. The problems include circle theorems, angles subtended by arcs, inscribed angles, and tangent-secant relationships.

Solve a series of geometry problems involving circle theorems, inscribed angles, tangent-secant relationships, and arc measures. Learn to apply key theorems such as the inscribed angle theorem and central angle theorem. Suitable for students in Grades 9-12.

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