Exterior Angles Of A Quadrilateral

Polygon is a closed, continued shape fabricated of straight lines. It may be a flat or a plane figure spanned beyond ii-dimensions. A polygon is an enclosed figure that can have more three sides. The lines forming the polygon are known equally the edges or sides and the points where they see are known as vertices. The sides that share a mutual vertex among them are known as adjacent sides. The angle enclosed inside the next side is chosen the interior bending and the outer bending is called the exterior angle.

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Exterior Angle

An exterior bending basically is formed by the intersection of any of the sides of a polygon and extension of the adjacent side of the chosen side. Interior and exterior angles formed within a pair of adjacent sides form a complete 180 degrees angle.

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Measures of Exterior Angles

They are formed on the outer part, that is, the exterior of the bending.
The respective sum of the exterior and interior bending formed on the aforementioned side = 180°.
The sum of all the exterior angles of the polygon is independent of the number of sides and is equal to 360 degrees, because it takes ane complete turn to cover polygon in either clockwise or anti-clockwise direction.
If we have a regular polygon of north sides, the measure of each exterior angle
= (Sum of all outside angles of polygon)/n
= (360 degree)/n

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Theorem for Exterior Angles Sum of a Polygon

If we detect a convex polygon, and then the sum of the exterior angle nowadays at each vertex will be 360°. Post-obit Theorem will explicate the exterior bending sum of a polygon:

Proof

Allow us consider a polygon which has north number of sides. The sum of the outside angles is N.

The sum of exterior angles of a polygon(Due north) =

Difference between {the sum of the linear pairs (180n)} – {the sum of the interior angles.(180(n – 2))}

North = 180n − 180(n – ii)
North = 180n − 180n + 360
N = 360

Hence, we have the sum of the exterior bending of a polygon is 360°.

Sample Problems on Outside Angles

Example one: Detect the exterior angle marked with 10.

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Solution:

Since the sum of exterior angles is 360 degrees, the post-obit properties hold:

∠1 + ∠two + ∠3 + ∠4 + ∠5 = 360°
l° + 75° + twoscore° + 125° + x = 360°
x = 360°

Example 2: Decide each outside bending of the quadrilateral.

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Solution:

Since, it is a regular polygon, measure of each exterior bending
= 360°
Number of sides
= 360°
four
= ninety°

Example 3: Find the regular polygon where each of the exterior bending is equivalent to 60 degrees.

Solution:

Since it is a regular polygon, the number of sides can be calculated past the sum of all outside angles, which is 360 degrees divided by the measure out of each outside angle.

Number of sides = Sum of all exterior angles of a polygon
n
Value of one pair of side = 360 degree
lx caste
= 6
Therefore, this is a polygon enclosed inside half dozen sides, that is hexagon.

Instance four: Find the interior angles 'x, y', and exterior angles 'w, z' of this polygon?

Exterior-Angles

Solution:

Here we have ∠DAC = 110° that is an exterior bending and ∠ACB = l° that is an interior angle.

Firstly we have to find interior angles 'x' and 'y'.
∠DAC + ∠x = 180° {Linear pairs}
110° + ∠10 = 180°
∠x = 180° – 110°
∠x = lxx°
At present,
∠x + ∠y + ∠ACB = 180° {Bending sum property of a triangle}
lxx°+ ∠y + 50° = 180°
∠y + 120° = 180°
∠y = 180° – 120°
∠y = lx°

Secondly now we can find outside angles 'w' and 'z'.
∠w + ∠ACB = 180° {Linear pairs}
∠w + l° = 180°
∠westward = 180° – l°
∠due west = 130°

Now nosotros can use the theorem exterior angles sum of a polygon,
∠west + ∠z + ∠DAC = 360° {Sum of exterior angle of a polygon is 360°}
130° + ∠z + 110° = 360°
240° + ∠z = 360°
∠z = 360° – 240°
∠z = 120°