Circle Theorem. Alternate segment.

Alternate Segment Theorem | Geometry Made Easy
In this video, we dive into the fascinating world of geometry with a focus on the **Alternate Segment Theorem**! This theorem is a powerful concept in circle geometry that connects the angles between a tangent and a chord with the angles in the opposite segment of the circle.

We’ll break down the theorem with step-by-step explanations, clear visuals, and real-life examples to help you fully understand and remember it. Whether you're a high school student, a geometry enthusiast, or just brushing up on your math skills, this video has everything you need to master the Alternate Segment Theorem.

**Topics Covered:**
- Definition of the Alternate Segment Theorem
- The relationship between tangent, chord, and angles in a circle
- How to apply the theorem in different geometric problems
- Practice examples and solutions

**Timestamps:**
00:00 - Introduction
01:20 - What is the Alternate Segment Theorem?
03:50 - Key Concepts & Terminology
06:15 - Step-by-Step Explanation
08:30 - Real-Life Examples
10:00 - Practice Problems
12:45 - Wrap-Up & Summary

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The Alternate Segment Theorem is a key result in circle geometry. It states:

**The angle between a tangent and a chord at the point of contact is equal to the angle in the alternate segment of the circle.**

### Understanding the Theorem
Imagine a circle with a tangent touching it at a point, say \( P \). Draw a chord from \( P \) to another point on the circle, creating a triangle inside the circle. This setup produces two segments of the circle on either side of the chord, with each segment containing an arc.

According to the theorem, the angle between the tangent line at \( P \) and the chord \( PQ \) (where \( Q \) is another point on the circle) is equal to the angle in the alternate segment (the segment not containing the arc where the chord lies).

### Steps to Prove or Apply the Theorem
1. Identify the tangent at point \( P \) and the chord \( PQ \).
2. Find the angle made by the tangent and chord at \( P \).
3. Locate the angle in the opposite or "alternate" segment of the circle from the chord \( PQ \).
4. According to the theorem, these two angles will be equal.

### Example Problem
Consider a circle with center \( O \), and let a tangent touch the circle at point \( P \). Suppose a chord \( PQ \) is drawn, creating two segments of the circle. The angle between the tangent at \( P \) and \( PQ \) will equal the angle inside the triangle formed by \( PQ \) and any other chord \( PR \) on the other side of the circle.

### Why This Theorem is Useful
The Alternate Segment Theorem is helpful in solving many geometric problems involving tangents, chords, and angles in circles, particularly in proving angle relationships within cyclic quadrilaterals and other configurations.

Let me know if you’d like a diagram or example worked out!

Видео Circle Theorem. Alternate segment. канала VINDAL'S ACADEMY