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Home / Geometry / Prove that the sum of the angles of a quadrilateral is equal to 4 right angles.

Prove that the sum of the angles of a quadrilateral is equal to 4 right angles.

Solution:
General enunciation: We have to prove that the sum of the angles of a quadrilateral is equal to 4 right angles.

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Particular enunciation: Let four angles of a quadrilateral ABCD be ∠A, ∠B, ∠C, and ∠D. We have to prove that sum of the four angles equal to 4 right angles that is ∠A+∠B+∠C+∠D= 4 right angles.

Construction: We join A with C and B with D.

Proof: We know the sum of the three angles of a triangles is equal to two right angles.

From triangle ABC

∠BAC+∠ABC+∠BCA=1800—————————————–(1)

Again from triangle ADC

∠DAC+∠ADC+∠DCA=1800——————————————(2)

Now adding (1) and (2) we have,

∠BAC+∠ABC+∠BCA+∠DAC+∠ADC+∠DCA=3600

⇒(∠BAC+∠DAC)+ ∠ABC+(∠BCA+∠DCA)+ ∠ADC=3600

⇒ ∠BAD+∠A BC+∠BCD+∠ADC=3600

⇒∠A+∠B+∠C+∠D=3600=4 right angles. (Proved)

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