Exterior angle are defined as the angles formed in between the next of the polygon and the extended surrounding side that the polygon. The exterior edge theorem says that once a triangle's side is extended, the result exterior angle developed is equal to the sum of the steps of the two opposite inner angles the the triangle. The theorem deserve to be provided to uncover the measure up of one unknown angle in a triangle. To use the theorem, we an initial need to recognize the exterior angle and then the connected two remote internal angles of the triangle.

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

What is Exterior angle Theorem?


The exterior edge theorem states that the measure up of an exterior edge is equal to the sum of the procedures of the two opposite(remote) internal angles of the triangle. Let united state recall a couple of common properties around the angles of a triangle: A triangle has 3 internal angles which constantly sum up to 180 degrees. It has actually 6 exterior angles and also this to organize gets applied to every of the exterior angles. Keep in mind that one exterior edge is supplementary come its nearby interior angle as they form a straight pair the angles.

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We have the right to verify the exterior angle theorem through the known 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 that angles)

Substituting the value of c in equation 3, us get

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

e = 180 - 180 + (a + b)

e = a + b

Hence verified.


Proof that Exterior edge Theorem


Consider a ΔABC. A, b and also c room the angle formed. Prolong the next BC to D. Now an exterior edge ∠ACD is formed. Attract a heat CE parallel come AB. Currently x and also y are the angles formed, where, ∠ACD = ∠x + ∠y

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

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


Exterior angle Inequality Theorem


The exterior edge inequality theorem states that the measure up of any kind of exterior angle of a triangle is greater than either of the opposite internal angles. This condition is solve by every the six external angles the a triangle.

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Exterior angle Theorem connected Articles

Check the end a couple of interesting posts related to Exterior edge Theorem.

Important notes

The exterior angle theorem claims that the measure up of one exterior edge is same to the amount of the measures of the 2 remote internal angles of the triangle.The exterior edge inequality theorem states that the measure of any exterior edge of a triangle is higher than one of two people of the opposite inner angles.The exterior angle and also the nearby interior angle are supplementary. All the exterior angles of a triangle amount up to 360º.

Solved Examples


Example 1: uncover the values of x and also y by using the exterior angle theorem that a triangle.

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Solution:

∠x is the exterior angle.

∠x + 92 = 180º (linear pair that 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 y space 88º and 47º respectively.


Example 2: discover BAC and 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: 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 amount 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 angle 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 edge Theorem


What is the Exterior angle Theorem?

The exterior edge theorem states that the measure of one exterior edge is equal to the amount of the actions of the two remote inner angles that the triangle. The remote interior angles are also called opposite inner angles.

How do you use the Exterior edge Theorem?

To usage the exterior edge theorem in a triangle we an initial need to identify the exterior angle and then the connected two remote internal angles that the triangle. A typical mistake of considering the adjacent interior angle have to be avoided. After identifying the exterior angles and also the related interior angles, we can apply the formula to find the absent angles or to develop a relationship between sides and also angles in a triangle.

What room Exterior Angles?

An exterior angle of a triangle is developed when any side of a triangle is extended. There are 6 exterior angles of a triangle as each of the 3 sides can be prolonged on both sides and 6 together exterior angles are formed.

What is the Exterior angle Inequality Theorem?

The measure of an exterior angle of a triangle is constantly greater 보다 the measure of either of the opposite interior angles that the triangle.

What is the Exterior edge Property?

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

What is the Exterior edge Theorem Formula?

The sum of the exterior angle = the sum of two non-adjacent inner opposite angles. One exterior edge of a triangle is equal to the sum of its two opposite non-adjacent internal angles.

Where should We use Exterior edge Theorem?

Exterior edge theorem can be supplied to identify the procedures 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 formed when a next of a polygon is extended. Every the exterior angle in every the polygons sum up come 360º.