Exterior angle are defined as the angle formed between the side of the polygon and the extended surrounding side the the polygon. The exterior edge theorem claims that as soon as a triangle's side is extended, the result exterior angle developed is same to the sum of the procedures of the 2 opposite inner angles of the triangle. The theorem can be offered to discover the measure of one unknown edge in a triangle. To apply the theorem, we first need to determine the exterior angle and then the linked two remote interior angles the the triangle.

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1.What is Exterior edge Theorem?
2.Proof of Exterior angle Theorem
3.Exterior angle Inequality Theorem
4.FAQs top top Exterior edge Theorem

What is Exterior angle Theorem?


The exterior angle theorem says that the measure up of one exterior edge is equal to the amount of the measures of the two opposite(remote) interior angles the the triangle. Let united state recall a few common properties about the angle of a triangle: A triangle has 3 inner angles which always sum approximately 180 degrees. It has actually 6 exterior angles and also this theorem gets used to every of the exterior angles. Note that one exterior angle is supplementary come its adjacent interior angle together they kind a direct pair of angles.

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We can verify the exterior edge theorem v the recognized properties that a triangle. Consider a Δ ABC.

The three angles a + b + c = 180 (angle sum property of a triangle) ----- Equation 1

c= 180 - (a+b) ----- Equation 2 (rewriting equation 1)

e = 180 - c----- Equation 3 (linear pair of angles)

Substituting the worth of c in equation 3, we get

e = 180 - <180 - (a+b)>

e = 180 - 180 + (a + b)

e = a + b

Hence verified.


Proof the Exterior angle Theorem


Consider a ΔABC. A, b and also c room the angles formed. Prolong the side BC come D. Currently an exterior angle ∠ACD is formed. Draw a line CE parallel to AB. Currently x and y space the angles formed, where, ∠ACD = ∠x + ∠y

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StatementReason
∠a = ∠xPair of alternate angles. (Since BA is parallel come CE and AC is the transversal).
∠b = ∠yPair of equivalent angles. (Since BA is parallel come CE and also BD is the transversal).
∠a + ∠b = ∠x + ∠yFrom the over statements
∠ACD = ∠x + ∠yFrom the building of CE
∠a + ∠b = ∠ACDFrom the over statements

Hence showed that the exterior edge of a triangle is same to the amount of the 2 opposite inner angles.


Exterior angle Inequality Theorem


The exterior angle inequality theorem says that the measure of any type of exterior angle of a triangle is higher than one of two people of the opposite interior angles. This condition is to solve by all the six external angles the a triangle.

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Exterior angle Theorem connected Articles

Check out a couple of interesting articles related come Exterior angle Theorem.

Important notes

The exterior edge theorem says that the measure up of one exterior edge is equal to the sum of the actions of the 2 remote inner angles that the triangle.The exterior edge inequality theorem claims that the measure up of any kind of exterior angle of a triangle is greater than one of two people of the opposite internal angles.The exterior angle and also the adjacent interior angle room supplementary. All the exterior angles of a triangle sum up to 360º.

Solved Examples


Example 1: uncover the worths of x and y by making use of the exterior edge theorem of a triangle.

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Solution:

∠x is the exterior angle.

∠x + 92 = 180º (linear pair of angles)

∠x = 180 - 92 = 88º

Applying the exterior edge theorem, us get, ∠y + 41 = 88

∠y = 88 - 41 = 47º

Therefore, the values of x and y are 88º and 47º respectively.


Example 2: find BAC and also ABC.

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Solution:

160º is an exterior angle of the Δ ABC. So, by using the exterior edge theorem, us have, ∠BAC + ∠ABC = 160º

x + 3x = 160º

4x = 160º

x = 40º

Therefore, ∠BAC = x = 40º and ∠ABC = 3xº = 120º


Example 3: find ∠ BAC, if ∠CAD = ∠ADC

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Solution:

Solving the straight pair in ~ vertex D, we gain ∠ADC + ∠ADE = 180º

∠ADC = 180º - 150º = 30º

Using the angle sum property, for Δ ACD,

∠ADC + ∠ACD + ∠CAD = 180º

∠ACD = 180 - ∠CAD -∠ADC

180º - ∠ADC -∠ADC (given ∠CAD= ∠ADC)

180º - 2∠ADC

180º - 2 × 30º

∠ACD = 180º - 60º = 120º

∠ACD is the exterior angle of ∠ABC

Using the exterior angle theorem, because that Δ ABC, ∠ACD = ∠ABC + ∠BAC

120º = 60º + ∠BAC

Therefore, ∠BAC = 120º - 60º = 60º.


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FAQs on Exterior angle Theorem


What is the Exterior edge Theorem?

The exterior edge theorem says that the measure of one exterior angle is equal to the amount of the actions of the 2 remote interior angles that the triangle. The remote internal angles are likewise called opposite internal angles.

How perform you use the Exterior angle Theorem?

To usage the exterior edge theorem in a triangle we very first need to recognize the exterior angle and also then the connected two remote interior angles that the triangle. A common mistake that considering the surrounding interior angle need to be avoided. After identify the exterior angles and also the related inner angles, we can apply the formula to uncover the lacking angles or to create a relationship in between sides and also angles in a triangle.

What space Exterior Angles?

An exterior angle of a triangle is developed when any side the a triangle is extended. There room 6 exterior angle of a triangle as each that the 3 sides deserve to be extended on both sides and also 6 such exterior angles room formed.

What is the Exterior edge Inequality Theorem?

The measure up of an exterior edge of a triangle is constantly greater than the measure of one of two people of the opposite interior angles of the triangle.

What is the Exterior angle Property?

An exterior edge of a triangle is equal to the amount of its 2 opposite non-adjacent interior angles. The sum of the exterior angle and the adjacent interior angle the is not opposite is equal to 180º.

What is the Exterior edge Theorem Formula?

The sum of the exterior edge = the sum of 2 non-adjacent internal opposite angles. One exterior angle of a triangle is same to the amount of its two opposite non-adjacent interior angles.

Where should We usage Exterior angle Theorem?

Exterior angle theorem can be offered to determine the steps of the unknown interior and also exterior angles of a triangle.

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Do all Polygons Exterior Angles add up to 360?

The exterior angle of a polygon are developed when a next of a polygon is extended. All the exterior angle in every the polygons amount up to 360º.