Question: kerjakan
1)
Tentukkan
\(\angle ACB\)!
2)
Tentukkan besar
\(\angle AOB\)!
3)
Tentukkan besar
\(\angle ADB\) dan \(\angle ACB\)!
1) Graph: circle diagram at top-left with points A, B, C on the circumference, center O inside, and a marked angle of 82° near the center; position_hint \in \{top-left\}.
2) Graph: circle diagram at top-center with points A, B, C on the circumference, center O inside, and intersecting chords/radii with a marked angle near 88° at the center-right; position_hint \in \{top-right\}.
3) Graph: circle diagram at top-right with points A, B, C, D on the circumference, center O inside, intersecting lines/chords, and a marked central angle near 88°; position_hint \in \{top-right\}.
1. **Problem 1: Find \(\angle ACB\) given the central angle \(\angle AOB = 82^\circ\).
2. **Problem 2: Find \(\angle AOB\) given an inscribed angle \(\angle ACB = 46^\circ\).
3. **Problem 3: Find \(\angle ADB\) and \(\angle ACB\) given \(\angle AOB = 80^\circ\).
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### Step 1: Understanding the circle angle relationships
- The central angle \(\angle AOB\) is the angle formed at the center \(O\) by points \(A\) and \(B\) on the circle.
- The inscribed angle \(\angle ACB\) is the angle formed on the circumference by points \(A\), \(C\), and \(B\).
- **Important rule:** The inscribed angle is half the measure of the central angle that subtends the same arc.
Mathematically,
$$\angle ACB = \frac{1}{2} \angle AOB$$
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### Problem 1: Find \(\angle ACB\) given \(\angle AOB = 82^\circ\)
1. Given \(\angle AOB = 82^\circ\).
2. Use the inscribed angle theorem:
$$\angle ACB = \frac{1}{2} \times 82^\circ$$
3. Calculate:
$$\angle ACB = 41^\circ$$
**Answer:** \(\angle ACB = 41^\circ\)
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### Problem 2: Find \(\angle AOB\) given \(\angle ACB = 46^\circ\)
1. Given \(\angle ACB = 46^\circ\).
2. Use the inscribed angle theorem rearranged:
$$\angle AOB = 2 \times \angle ACB$$
3. Calculate:
$$\angle AOB = 2 \times 46^\circ = 92^\circ$$
**Answer:** \(\angle AOB = 92^\circ\)
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### Problem 3: Find \(\angle ADB\) and \(\angle ACB\) given \(\angle AOB = 80^\circ\)
1. Given \(\angle AOB = 80^\circ\).
2. By the inscribed angle theorem:
$$\angle ACB = \frac{1}{2} \times 80^\circ = 40^\circ$$
3. For \(\angle ADB\), if points \(A, D, B\) lie on the circle and \(D\) is on the arc opposite to \(C\), then \(\angle ADB\) also subtends the same arc as \(\angle ACB\), so:
$$\angle ADB = \angle ACB = 40^\circ$$
**Answer:** \(\angle ADB = 40^\circ\) and \(\angle ACB = 40^\circ\)
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### Summary of answers:
- Problem 1: \(\angle ACB = 41^\circ\)
- Problem 2: \(\angle AOB = 92^\circ\)
- Problem 3: \(\angle ADB = 40^\circ\), \(\angle ACB = 40^\circ\)