THEOREM TITLE:
(Exterior angle sum property) If the sides of a quadrilateral are produced in order, the sum of four exterior angles so formed is 360º.
Proof: Let the sides of a quadrilateral be produced in order as shown in figure, forming exterior angles ∠1, ∠2, ∠3 and ∠4. Since ∠1 and ∠ form a linear pair and the sum of the angles of a linear pair is 180º. ∴∠1 + ∠A = 180º ----(i) Similarly, we have ∠2 + ∠B = 180º ----(ii) ∠3 + ∠C = 180º ----(iii) and, ∠4 + ∠D = 180º ----(iv)
→ ∠1 + ∠2 + ∠3 + ∠4 = 720º - 360º = 360º
THEOREM TITLE:
The sum of all the exterior angles formed by producing the sides of a convex polygon in the same order is equal to four right angles.
Given: A convex polygon P1 P2 P3 P4 P5.Its sides P1 P2,P2P3,P3P4, P5P1are produced in order, forming exterior angles ∠1, ∠2, ∠3, ∠4 and ∠5.
To Prove: ∠1 + ∠2 + ∠3 + ∠4 + ∠5 = 4 right angles.
Construction: Take any point O, outside the polygon. Draw OA1,OA2,OA3,OA4, and OA5 parallel to and in the same sense as P1P2,P3P4,P4 P5 and P4 P5, and P5 P1 respectively.
Proof: Since the arms of ∠ and ∠a are parallel and drawn in the same sense. ∴ ∠1 = ∠a Similarly,∠2 = ∠b, ∠3 = ∠c,∠4 = ∠d and ∠5 = ∠e ∴ ∠1 + ∠2 + ∠3 + ∠4 + ∠5 = ∠a + ∠b + ∠c + ∠d + ∠e = 360º [Sum of the angles at a point is 360º] = 4 x 90º
= 4 right angles.
| ➲Each exterior angle of a regular polygon of n sides is equal to ➲If there is a regular polygon of n sides (n ≥ 3), then its each interior angle is equal to |
Answer
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The sides of a quadrilateral are produced in order. What is the sum of the four exterior angles?
Solution
Let ABCD be a quadrilateral such that exterior angles formed by extending the sides AD,AB,BC and CD are angles x,y,z and w.
Now, since, sides AD,AB,BC and CD are straight lines, therefore,
⇒∠BAD+x=180∘⇒∠BAD=180∘−x...(i) [Linear Pair].
Similarly,
⇒∠ABC=180∘−y ...(ii)
⇒∠BCD=180∘−z ...(iii)
⇒∠ADB=180∘−w ...(iv)
Adding (i), (ii), (iii) and (iv), we get,
⇒∠BAD+∠ABC+∠BCD+∠ADB=(180∘−x)+(180∘−y)+(180∘−z)+(180∘−w)
⇒360∘=720∘−(x+y+z+w) [Sum of all the angles of a quadrilateral is 360∘]
⇒x+y+z+w=360∘
Thus, the sum of the four exterior angles is 360∘.
OR
It is well established that regardless of number of exterior angles, the sum of all the exterior angles of a polygon is always 360∘.
Mathematics
RD Sharma
Standard VIII
2