Question

Prove that the sum of the angles of a quadrilateral is 360°.​

Answer 1

Answer:

We know that the sum of the angles of a triangle is 180°. ⇒ ∠A + ∠B + ∠C + ∠D = 360° [using (i) and (ii)]. Hence, the sum of all the four angles of a quadrilateral is 360°.

Step-by-step explanation:

If we draw one diagonal to quadrilateral we will get two triangles i.e. △ABC and △ACD and we know that sum of the angles of a triangle is 180

o

Sum of angles of quadrilateral ABCD=sum of angles of △ABC + sum of the angles △ACD

= 180

o

+180

o

= 360

o

Answer 2

Answer:

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.

Hope it help you my friend