📐 Angle Chasing
Angle chasing is the systematic process of finding unknown angles using known relationships. It’s one of the most common problem types in AMC geometry.
🎯 Key Angle Relationships
Triangle Angle Sum
Fundamental: The sum of angles in any triangle is $180°$.
Applications:
- Find third angle when two are known
- Prove angles are equal by showing they’re both $180° - \text{other angles}$
- Use in angle chasing chains
Parallel Line Theorems
When two lines are parallel:
| Relationship | Description | Example |
|---|---|---|
| Corresponding | Angles in same position | $\angle 1 = \angle 5$ |
| Alternate Interior | Angles on opposite sides, between lines | $\angle 3 = \angle 6$ |
| Same-Side Interior | Angles on same side, between lines | $\angle 3 + \angle 5 = 180°$ |
| Alternate Exterior | Angles on opposite sides, outside lines | $\angle 1 = \angle 7$ |
Exterior Angle Theorem
Key Insight: An exterior angle equals the sum of the two non-adjacent interior angles.
Formula: $\angle ACD = \angle A + \angle B$ (where $D$ is on extension of $BC$)
Inscribed Angle Theorem
Fundamental: An inscribed angle is half the measure of its intercepted arc.
Special Cases:
- Angle inscribed in semicircle is right angle
- Angles inscribed in same arc are equal
- Opposite angles in cyclic quadrilateral sum to $180°$
🔍 Micro-Examples
Triangle Angle Sum
In $\triangle ABC$, if $\angle A = 50°$ and $\angle B = 70°$, then $\angle C = 180° - 50° - 70° = 60°$.
Parallel Lines
If $AB \parallel CD$ and $\angle 1 = 40°$, then $\angle 2 = 40°$ (corresponding angles).
Exterior Angle
In $\triangle ABC$, if $\angle A = 30°$ and $\angle B = 50°$, then exterior angle at $C$ is $30° + 50° = 80°$.
Inscribed Angle
If arc $AB$ measures $100°$, then inscribed angle $\angle ACB$ measures $50°$.
⚠️ Common Traps
Pitfall: Assuming parallel lines without proof
- Fix: Look for equal corresponding angles or use given information
Pitfall: Confusing inscribed and central angles
- Fix: Inscribed angle is half the arc, central angle equals the arc
Pitfall: Wrong angle sum in polygons
- Fix: Triangle sum is $180°$, quadrilateral sum is $360°$
Pitfall: Forgetting about supplementary angles
- Fix: Linear pairs sum to $180°$, same-side interior angles sum to $180°$
🎯 AMC-Style Worked Example
Problem: In the figure, $AB \parallel CD$, $\angle BAE = 40°$, and $\angle DCE = 60°$. Find $\angle AEC$.
Solution: Since $AB \parallel CD$, we can use parallel line properties.
Let’s draw auxiliary line $EF \parallel AB \parallel CD$ through point $E$.
Now we have:
- $\angle BAE = \angle AEF = 40°$ (alternate interior angles)
- $\angle DCE = \angle CEF = 60°$ (alternate interior angles)
Therefore, $\angle AEC = \angle AEF + \angle CEF = 40° + 60° = 100°$.
Answer: $\angle AEC = 100°$
🔗 Related Topics
- Triangles Basics - Triangle angle properties
- Circles & Power of Point - Inscribed angle applications
- Cyclic Quadrilaterals - Opposite angle relationships
- Similarity & Ratios - Using angles in similarity
💡 Quick Reference
Angle Chasing Strategy
- Mark given angles on the diagram
- Look for parallel lines - they create equal angles
- Check for triangles - use angle sum theorem
- Look for circles - use inscribed angle theorem
- Follow the chain - each angle leads to the next
Common Patterns
- Parallel lines: Create equal corresponding/alternate angles
- Triangles: Use angle sum to find missing angles
- Circles: Inscribed angles are half the arc measure
- Linear pairs: Sum to $180°$
Special Angles
- Right angles: $90°$
- Straight angles: $180°$
- Complete rotation: $360°$
- Equilateral triangle: All angles $60°$
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