https://r2.erweima.ai/mathbot/image/1726726818812of9olomi.jpg

The image shows a circle with several points on its circumference and one interior point, \(O\), that is the center of the circle. Points \(A\), \(B\), \(C\), \(D\), and \(E\) are on the circumference, and various angles are formed between them. The problem states that:

\[
\angle ACB + \angle ADB + \angle AEB = 90^\circ
\]
and asks for the measure of \(\angle AOB\).

### Approach:
1. Points \(A\), \(B\), and \(O\) lie on the circle, and \(\angle AOB\) is the central angle.
2. The sum of the given three angles (i.e., \(\angle ACB\), \(\angle ADB\), and \(\angle AEB\)) is provided as \(90^\circ\).
3. We will use the fact that the central angle is twice the inscribed angle subtended by the same arc. This is a key property of circles.

#### Step-by-step Solution:
- Since the sum of the inscribed angles is given as \(90^\circ\), we know that these angles together subtend the arc \(AB\).
- According to the circle theorem, the central angle subtended by the same arc is double the sum of the inscribed angles.

Thus:
\[
\angle AOB = 2 \times (90^\circ) = 180^\circ
\]

So, the measure of \(\angle AOB\) is \(180^\circ\).

---

Do you have any further questions or would you like more details?

### Related Questions:
1. What is the circle theorem that connects central and inscribed angles?
2. How can you prove that \(\angle AOB = 2 \times \) (sum of inscribed angles)?
3. Why is the sum of inscribed angles subtended by the same arc a constant value?
4. How can you apply the same logic to find other central angles?
5. What happens if the sum of the inscribed angles is greater than \(90^\circ\)?

**Tip**: Remember that the central angle is always double any inscribed angle subtended by the same arc!

Perhatikan gambar berikut ini. \(\angle ACB + \angle ADB + \angle AEB = 90^\circ\). Maka besar \(\angle AOB\) adalah ....

Learn how to find the central angle \(\angle AOB\) in a circle where the sum of the inscribed angles \(\angle ACB + \angle ADB + \angle AEB = 90^\circ\). This geometry problem involves circle theorems, particularly the central angle theorem.

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