sp the perpendicular line = 180 degrees and the line subtendinf that is 90 degrees/

Let's clarify the concept involving angles and perpendicular lines within the context of circles.

### Understanding Perpendiculars and Subtended Angles:

1. **Perpendicular Line and 180 Degrees**:
   - If a line (or a chord) is perpendicular to the diameter of the circle and intersects the circle at two points, it indeed creates two right angles (90 degrees each) at those points.
   - However, when we say a line is 180 degrees, we're usually referring to a straight line, which is a semicircle (half of the circle).

2. **Angle Subtended by the Diameter**:
   - If a line (like the diameter) subtends an angle at the circumference, this angle will always be \(90^\circ\). This is a key property of circles, known as **Thales' Theorem**: any angle subtended by a semicircle (or diameter) on the circumference is a right angle.

### Application to Your Case:
- If the line you referred to as perpendicular is in fact the diameter of the circle, then any angle formed at the circumference by this line will be \(90^\circ\). 
- Therefore, the angle \(a\) that subtends the arc from the center is \(60^\circ\), and half of it is \(30^\circ\) as previously mentioned.
- If there’s a perpendicular line from the center to a chord, it would indeed create two \(90^\circ\) angles where it intersects with the circumference.

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Would you like further clarification on any part of this explanation? Here are some follow-up questions to consider:

1. What is the relationship between a diameter and angles subtended on the circle?
2. How does Thales' Theorem apply to various circle-based problems?
3. What happens to angles subtended by the same arc at different points on the circumference?
4. How does the perpendicular from the center of a circle to a chord relate to the chord?
5. Can we use these properties to solve problems involving chords and arcs?

**Tip:** When dealing with circles, always remember that angles subtended by the same arc at the circumference are equal, and the diameter subtends a \(90^\circ\) angle at any point on the circle.

Explore the concept of perpendicular lines and subtended angles in circle geometry. Learn how a line perpendicular to the diameter of a circle creates two right angles (90 degrees each) at its intersections with the circle's circumference. Understand Thales' Theorem, which states that any angle subtended by a semicircle (or diameter) on the circumference is a right angle. This explanation clarifies the relationship between diameters, angles subtended on circles, and their applications in problem-solving.

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