Exterior angles are identified as the angles formed between the side of the polygon and the extended nearby side of the polygon. The exterior edge theorem claims that when a triangle's side is extended, the result exterior angle formed is same to the sum of the procedures of the 2 opposite inner angles that the triangle. The theorem can be provided to uncover the measure up of one unknown edge in a triangle. To use the theorem, we very first need to recognize the exterior angle and also then the connected two remote inner angles the the triangle.

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1.What is Exterior edge Theorem?
2.Proof the Exterior edge Theorem
3.Exterior edge Inequality Theorem
4.FAQs ~ above Exterior edge Theorem

What is Exterior angle Theorem?


The exterior angle theorem says that the measure of an exterior edge is equal to the sum of the steps of the 2 opposite(remote) inner angles the the triangle. Let united state recall a few common properties about the angle of a triangle: A triangle has actually 3 interior angles which constantly sum approximately 180 degrees. It has actually 6 exterior angles and this organize gets used to each of the exterior angles. Keep in mind that one exterior edge is supplementary to its surrounding interior angle as they form a straight pair the angles.

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We deserve to verify the exterior angle theorem with the well-known properties of a triangle. Take into consideration a Δ ABC.

The 3 angles a + b + c = 180 (angle sum building of a triangle) ----- Equation 1

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

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

Substituting the value of c in equation 3, we get

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

e = 180 - 180 + (a + b)

e = a + b

Hence verified.


Proof that Exterior angle Theorem


Consider a ΔABC. A, b and also c space the angles formed. Extend 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 angle formed, where, ∠ACD = ∠x + ∠y

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StatementReason
∠a = ∠xPair of alternating angles. (Since BA is parallel to CE and also AC is the transversal).
∠b = ∠yPair of matching angles. (Since BA is parallel to CE and BD is the transversal).
∠a + ∠b = ∠x + ∠yFrom the above statements
∠ACD = ∠x + ∠yFrom the building and construction of CE
∠a + ∠b = ∠ACDFrom the over statements

Hence verified that the exterior angle of a triangle is equal to the sum of the two opposite internal angles.


Exterior edge Inequality Theorem


The exterior angle inequality theorem states that the measure up of any kind of exterior edge of a triangle is greater than one of two people of the opposite interior angles. This problem is satisfied by all the six outside angles the a triangle.

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Exterior angle Theorem associated Articles

Check the end a couple of interesting write-ups related to Exterior angle Theorem.

Important notes

The exterior angle theorem says that the measure of an exterior edge is same to the sum of the steps of the two remote interior angles that the triangle.The exterior edge inequality theorem states that the measure up of any exterior angle of a triangle is greater than either of the opposite inner angles.The exterior angle and the surrounding interior angle are supplementary. Every the exterior angle of a triangle sum up to 360º.

Solved Examples


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

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Solution:

∠x is the exterior angle.

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

∠x = 180 - 92 = 88º

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

∠y = 88 - 41 = 47º

Therefore, the worths of x and also y room 88º and 47º respectively.


Example 2: find BAC and ABC.

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Solution:

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

x + 3x = 160º

4x = 160º

x = 40º

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


Example 3: uncover ∠ BAC, if ∠CAD = ∠ADC

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Solution:

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

∠ADC = 180º - 150º = 30º

Using the angle sum property, because that 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 edge of ∠ABC

Using the exterior edge 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 angle Theorem?

The exterior angle theorem says that the measure up of an exterior edge is same to the sum of the measures of the two remote inner angles that the triangle. The remote internal angles are also called opposite inner angles.

How carry out you usage the Exterior angle Theorem?

To use the exterior angle theorem in a triangle we an initial need to identify the exterior angle and also then the associated two remote inner angles the the triangle. A usual mistake the considering the nearby interior angle have to be avoided. After identifying the exterior angles and also the related interior angles, we can use the formula to discover the missing angles or to develop a relationship between sides and also angles in a triangle.

What space Exterior Angles?

An exterior edge of a triangle is created when any type of side of a triangle is extended. There are 6 exterior angles of a triangle as each that the 3 sides have the right to be extended on both sides and 6 such exterior angles are formed.

What is the Exterior angle Inequality Theorem?

The measure of one exterior angle of a triangle is constantly greater 보다 the measure of one of two people of the opposite internal angles the the triangle.

What is the Exterior angle Property?

An exterior edge of a triangle is same to the sum of its two opposite non-adjacent interior angles. The sum of the exterior angle and the nearby interior angle that is not opposite is same to 180º.

What is the Exterior angle Theorem Formula?

The sum of the exterior angle = the sum of two non-adjacent inner opposite angles. An exterior edge of a triangle is same to the sum of its 2 opposite non-adjacent inner angles.

Where need to We usage Exterior edge Theorem?

Exterior edge theorem might be supplied to identify the steps of the unknown interior and also exterior angles of a triangle.

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Do every Polygons Exterior Angles add up to 360?

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