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Angle Sum Property of a Quadrilateral

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Sum of Four Angles of a Quadrilateral:

  • Cut out a paper in the shape of a quadrilateral.
  • Make folds in it that join the vertices of opposite angles.

  • Take two triangular pieces of paper such that one side of one triangle is equal to one side of the other.
  • Let us suppose that in ∆ABC and ∆PQR, sides AC and PQ are the equal sides.

  • Join the triangles so that their equal sides lie side by side. 
  • We used two triangles to obtain a quadrilateral. The sum of the three angles of a triangle is 180°.
  • Hence, The sum of the measures of the four angles of a quadrilateral is 360°.

theorem

Angle Sum Property of a Quadrilateral:

Theorem: The sum of the angles of a quadrilateral is 360°.

Construction: This can be verified by drawing a diagonal AC and dividing the quadrilateral into two triangles.

Proof:

Let ABCD be a quadrilateral and AC be diagonal.

In △ ABC,

You know that,
∠ B + ∠ BAC + ∠ BCA = 180°........(1)
Similarly, in △ADC,
∠ D + ∠ DAC + ∠ DCA = 180°........(2)
Adding (1) and (2), we get,
∠ B + ∠ BAC + ∠ BCA + ∠ D + ∠ DAC + ∠ DCA = 180° + 180°

Also, ∠ BAC + ∠ DAC = ∠ A and ∠ BCA + ∠ DCA = ∠ C

So, ∠ A + ∠ B + ∠ C + ∠ D = 180° + 180°= 360°

i.e., The sum of the angles of a quadrilateral is 360°.

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Shaalaa.com | Prove Sum of Interior Angles of a Quadrilateral are 360 Degrees

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