Scissor Theorem in Geometry – Concept & Practice Questions
June 4, 2024 2025-06-04 15:21Scissor Theorem in Geometry – Concept & Practice Questions
Scissor Theorem in Geometry – Concept & Practice Questions
Amiya Kumar ✂️ Scissor Theorem in Geometry – Concept & Practice Questions
What is the Scissor Theorem?
In the given figure, the triangle is "cut" by an internal triangle in such a way that: \[ \angle D = \angle A + \angle B + \angle C \] This geometric property is popularly referred to as the Scissor Theorem because of the scissors-like structure formed.
Diagram showing internal angle D = A + B + C
🔎 Alternate Visualization – With External Angles
Using an auxiliary construction, you can extend the side and interpret \( A + B \) as an exterior angle, helping reinforce the theorem geometrically.
Extension showing \( A + B = \angle E \)
🧠 Practice Questions
Given: \( \angle B = 10^\circ, \angle C = 20^\circ, \angle D = 80^\circ \)
Solution:
By Scissor Theorem: \[ \angle D = \angle A + \angle B + \angle C \Rightarrow 80 = A + 10 + 20 \Rightarrow A = 50^\circ \]
Solution:
\[ \angle D = A + B + C = 40 + 40 + 40 = \boxed{120^\circ} \]
Solution:
\[ \angle D = A + B + C \Rightarrow 2x = x + 20 + 30 \Rightarrow x = 50^\circ \Rightarrow A = x = 50^\circ \]
🎯 Conclusion
The Scissor Theorem is a powerful geometric insight that’s often used in Olympiads and entrance exams to test understanding of angle relations in a triangle. Always look for symmetric patterns and internal triangles—many surprises are hidden in the angles!
Keep cutting through confusion—one triangle at a time! ✂️📐
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