Question

Draw a quadrilateral and mark any two outer angles. Is
there any relation between the sum of these two and
the inner angles at the other two vertices?​

Answer 1

Answer:

Step-by-step explanation:

Answer :

∠ ADC + ∠ CDF = 180° (linear pair of angles at a vertex)

∠ CDF = 180° - ∠ ADC …(1)

∠ ABC + ∠ CBE = 180° (linear pair of angles at a vertex)

∠ CBE = 180° - ∠ ABC … (2)

Sum of two exterior angles marked.

⇒ ∠ CBE + ∠ CDF = 180° - ∠ ABC + 180° - ∠ ADC

⇒ ∠ CBE + ∠ CDF = 360° - (∠ ABC + ∠ ADC) …(3)

In ABCD

∠ ABC + ∠ BCD + ∠ ADC + ∠ DAB = 360°

[sum of all interior angles 4-sided polygon is 360°]

⇒ ∠ ABC + ∠ ADC = 360° - ∠ BCD - ∠ DAB

Put this value in equation (3)

⇒ ∠ CBE + ∠ CDF = 360° - (360° - ∠ BCD - ∠ DAB)

⇒ ∠ CBE + ∠ CDF = 360° - 360° + ∠ BCD + ∠ DAB

⇒ ∠ CBE + ∠ CDF = ∠ BCD + ∠ DAB

Hence, yes there is a relation between the sum of exterior angles marked and sum of inner angles at the other two vertices.