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Proof: Base Angles of an Isosceles Triangle Are Equal (Without Auxiliary Lines)
Theorem
In an isosceles triangle with , we must prove that:
Proof Using Rotation
1. Rotate the Triangle
• Consider rotating by 180° around point .
• Since , this rotation moves point B to point C and point C to point B.
2. Corresponding Side Relationships
• Rotation preserves distances, so the rotated triangle is identical to the original.
• The side coincides with , and side remains unchanged.
3. Corresponding Angle Relationships
• Rotation preserves angles, meaning maps directly onto .
• This gives us the equality:
Conclusion
Since the triangle remains unchanged under a 180° rotation, the base angles of an isosceles triangle must be equal.
Q.E.D.
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