Prove the sum of intern angles of a quadrilateral is equal to 360 ∘ You can use the information: theorem "The sum of all angles in a triangle is equal to 180 º " to prove the sum of all angles in a quadrilateral is 360 ∘ .

Question

Prove the sum of intern angles of a quadrilateral is equal to       360          ∘      You can use the information: theorem &quot;The sum of all angles in a triangle is equal to   180      º  &quot; to prove the sum of all angles in a quadrilateral is       360          ∘      .

Bernard Scott · Accepted Answer

Step 1Call the angles of the quadrilateral A, B, C, D, with A opposite of D and B opposite of C. Our goal is to find   ∠  A  +  ∠  B  +  ∠  C  +  ∠  D.Create the line BD. We now have two triangles,   △  A  B  D and   △  C  B  D.We have   ∠  B  =  ∠  A  B  D  +  ∠  C  B  D     and     ∠  D  =  ∠  A  D  B  +  ∠  C  D  B  ..Step 2Moreover, since we have triangles, we know that  ∠  A  +  ∠  A  B  D  +  ∠  A  D  B  =      180    ∘       and     ∠  C  +  ∠  C  B  D  +  ∠  C  D  B  =      180    ∘  Therefore,                     ∠        A        +        ∠        B        +        ∠        C        +        ∠        D                            =        ∠        A        +        (        ∠        A        B        D        +        ∠        C        B        D        )        +        ∠        C        +        (        ∠        A        D        B        +        ∠        C        D        B        )                                          =        [        ∠        A        +        ∠        A        B        D        +        ∠        A        D        B        ]        +        [        ∠        C        +        ∠        C        B        D        +        ∠        C        D        B        ]                                          =                  180          ∘                +                  180          ∘                                                  =                  360          ∘

Prove the sum of intern angles of a quadrilateral is equal to 360^circ

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