Quadrilateral Theorems

 

ABOUT:

Proof: Let ABCD be a quadrilateral. Join AC. 

Clearly, ∠1 + ∠2 = ∠A ...... (i) 

And, ∠3 + ∠4 = ∠C ...... (ii) 

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

Therefore, from ∆ABC, we have

∠2 + ∠4 + ∠B = 180° (Angle sum property of triangle)

From ∆ACD, we have 

∠1 + ∠3 + ∠D = 180° (Angle sum property of triangle) 

Adding the angles on either side, we get; 

∠2 + ∠4 + ∠B + ∠1 + ∠3 + ∠D = 360° 

⇒ (∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360° 

⇒ ∠A + ∠B + ∠C + ∠D = 360° [using (i) and (ii)]. 

Hence, the sum of all the four angles of a quadrilateral is 360°.

(OR)

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

Proof: Let ABCD be a quadrilateral. Join AC. 

Clearly, ∠1 + ∠2 = ∠A ...... (i) 

And, ∠3 + ∠4 = ∠C ...... (ii) 

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


From ∆ACD, we have 

∠1 + ∠3 + ∠D = 180° (Angle sum property of triangle) 

Adding the angles on either side, we get; 

∠2 + ∠4 + ∠B + ∠1 + ∠3 + ∠D = 360° 

⇒ (∠1 + ∠2) + ∠B + (∠3 + ∠4) + ∠D = 360° 

⇒ ∠A + ∠B + ∠C + ∠D = 360° [using (i) and (ii)]

Hence, the sum of all the four angles of a quadrilateral is 360°.

DEMONSTRATION:

  1. The angle of a quadrilateral are (3x + 2)°, (x – 3), (2x + 1)°, 2(2x + 5)° respectively. Find the value of x and the measure of each angle.

    Solution:            

    Using angle sum property of quadrilateral, we get

     (3x + 2)°+ (x – 3)° + (2x + 1)° +  2(2x + 5)°= 360°                 

    ⇒ 3x + 2 + x - 3 + 2x + 1 + 4x + 10 = 360°

    ⇒ 10x + 10 = 360                              

    ⇒ 10x = 360 – 10                              

    ⇒ 10x = 350                       

    ⇒ x = 350/10                     

    ⇒ x = 35                                              

    Therefore, (3x + 2) = 3 × 35 + 2 = 105 + 2 = 107°

    (x – 3) = 35 – 3 = 32°

    (2x + 1) = 2 × 35 + 1 = 70 + 1 = 71°

    2(2x + 5) = 2(2 × 35 + 5) = 2(70 + 5) = 2 × 75 = 150°

    Therefore, the four angles of the quadrilateral are 32°, 71° 107°, 150° respectively.