Question

Prove angle sum property in a quadrilateral.​

Answer 1

Answer:

According to the angle sum property of a Quadrilateral, the sum of all the four interior angles is 360 degrees.

Quadrilateral: Angle Sum Property

Proof: In the quadrilateral ABCD,

∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles.

AC is a diagonal

AC divides the quadrilateral into two triangles, ∆ABC and ∆ADC

We have learned that the sum of internal angles of a quadrilateral is 360°, that is, ∠ABC + ∠BCD + ∠CDA + ∠DAB = 360°.

let’s prove that the sum of all the four angles of a quadrilateral is 360 degrees.

We know that the sum of angles in a triangle is 180°.

Now consider triangle ADC,

∠D + ∠DAC + ∠DCA = 180° (Sum of angles in a triangle)

Now consider triangle ABC,

∠B + ∠BAC + ∠BCA = 180° (Sum of angles in a triangle)

On adding both the equations obtained above we have,

(∠D + ∠DAC + ∠DCA) + (∠B + ∠BAC + ∠BCA) = 180° + 180°

∠D + (∠DAC + ∠BAC) + (∠BCA + ∠DCA) + ∠B = 360°

We see that (∠DAC + ∠BAC) = ∠DAB and (∠BCA + ∠DCA) = ∠BCD.

Replacing them we have,

∠D + ∠DAB + ∠BCD + ∠B = 360°

That is,

∠D + ∠A + ∠C + ∠B = 360°.

Or, the sum of angles of a quadrilateral is 360°. This is the angle sum property of quadrilaterals.

Answer 2

According to the angle sum property of a Quadrilateral, the sum of all the four interior angles is 360 degrees. Proof: In the quadrilateral ABCD, ∠ABC, ∠BCD, ∠CDA, and ∠DAB are the internal angles. AC is a diagonal.

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