Draw a quadrilateral $A B C D .$ Measure the four angles $∠ A, ∠ B, ∠ C$ and $∠ D .$ Verify whether their sum is $360^{\circ} .$

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Solution

Consider a quadrilateral $P Q R S$
join $Q S$
now,
$\begin{matrix} C o n s i d e r t r i a n g l e P Q S, w e h a v e ∠ P + ∠ P Q S + ∠ P S Q = 180^{\circ} - - - (1) s i m i l a r l y, i n t r i a n g l e Q R S, w e h a v e ∠ S Q R + ∠ R + ∠ Q S R = 180^{\circ} - - - - (2) O n a d d i n g (1) a n d (2) w e g e t ∠ P + ∠ P Q R + ∠ P S Q + ∠ S Q R + ∠ R + ∠ Q S R = 180^{\circ} + 180^{\circ} ∠ P + ∠ Q + ∠ R + ∠ S = 360^{\circ} p o v e d, \end{matrix}$

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Draw a quadrilateral ABCD. Measure the four angles ∠A,∠B,∠C and ∠D. Verify whether their sum is 360∘.

Consider a quadrilateral PQRSjoin QSnow,ConsidertrianglePQS,wehave∠P+∠PQS+∠PSQ=180∘−−−(1)similarly,intriangleQRS,wehave∠SQR+∠R+∠QSR=180∘−−−−(2)Onadding(1)and(2)weget∠P+∠PQR+∠PSQ+∠SQR+∠R+∠QSR=180∘+180∘∠P+∠Q+∠R+∠S=360∘poved,

Draw a quadrilateral $A B C D .$ Measure the four angles $∠ A, ∠ B, ∠ C$ and $∠ D .$ Verify whether their sum is $360^{\circ} .$