Most Important Proof Question from Quadrilateral Chapter in Class 9th

Proof questions play a crucial role in understanding the properties and relationships in quadrilaterals. One of the most important proof questions in the quadrilateral chapter for Class 9 is the proof of the sum of all angles in a quadrilateral.

Proof of the Sum of all Angles in a Quadrilateral:

- Start by considering a quadrilateral ABCD.
- Draw a diagonal AC which divides the quadrilateral into two triangles, namely triangle ABC and triangle CDA.
- In triangle ABC, the sum of angles is 180 degrees (as in any triangle).
- Therefore, angle ABC + angle BAC + angle ACB = 180 degrees.
- Similarly, in triangle CDA, the sum of angles is also 180 degrees.
- So, angle CDA + angle DAC + angle ACD = 180 degrees.
- Adding the equations from both triangles, we get:
(angle ABC + angle BAC + angle ACB) + (angle CDA + angle DAC + angle ACD) = 180 + 180
angle ABC + angle BAC + angle ACB + angle CDA + angle DAC + angle ACD = 360 degrees
- This equation represents the sum of all angles in a quadrilateral ABCD, which is 360 degrees.
- Hence, the proof is complete, and it establishes that the sum of all angles in a quadrilateral is always 360 degrees.

Understanding and mastering this proof question is essential for students to grasp the concept of quadrilaterals thoroughly. It also helps in building a strong foundation for further geometrical concepts.