In Circle M, angle AMB is 74 degrees. What is angle CDB
---**Question 1:**
1. An inscribed angle is ____ the measu [20]
**Question 2:**
2. In circle M, m∠AMB = 74°. What is m∠CDB? [21]
**Diagram Description (for Question 2):**
* **Type:** Geometric figure - Circle.
* **Main Elements:**
* A circle is shown.
* The center of the circle is labeled 'M'.
* Four points are located on the circumference of the circle: A, B, C, and D.
* Line segments connect the center M to points A and B (radii).
* Line segments connect points on the circumference: A to B (chord), C to B (chord), C to D (chord), and A to D (chord).
* Angle ∠AMB is formed at the center M by radii MA and MB. This angle is labeled as 74°.
* Angle ∠CDB is an inscribed angle formed by chords CD and DB, with vertex D on the circumference. This is the angle whose measure is requested.
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We have circle M with center M. The central angle AMB measures 74 degrees. We need to find the inscribed angle CDB. Let's examine the relationship between these two types of angles.
A fundamental theorem states that a central angle equals the measure of its intercepted arc. Since the central angle AMB measures 74 degrees, the arc AB also measures 74 degrees. This arc measurement is crucial for our next step.
Now we apply the inscribed angle theorem. This theorem states that an inscribed angle is half the measure of its intercepted arc. The inscribed angle CDB intercepts the same arc AB that measures 74 degrees. Therefore, angle CDB equals one half times 74 degrees.
Let's complete the calculation. We have angle CDB equals one half times 74 degrees. One half times 74 equals 37. Therefore, the inscribed angle CDB measures 37 degrees. This demonstrates the inscribed angle theorem in action.
To summarize: A central angle equals its intercepted arc measure. An inscribed angle is half its intercepted arc measure. The central angle AMB of 74 degrees creates an arc AB of 74 degrees. The inscribed angle CDB equals 37 degrees, which is half of the arc AB. This fundamental relationship applies to all circles.