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Home >> Triangle >> Properties >> Exterior Angle Property of a Triangle >>

Exterior Angle Property of a Triangle

Sum of Two Sides Angle Sum Property Angles opposite to equal sides of triangle are equal Angle opposite to longer side is greater Pythagoras Theorem
Exterior Angle Property of a Triangle Mid point property of Triangle Triangles on same base & between same parallel lines

Before you study exterior angle property of a triangle, you are advised to read:

What are Interior Angles of a Triangle ?
What are Exterior Angles of a Triangle
What are Interior Opposite Angles ?

Exterior Angle Property of a Triangle states that measure of exterior angle of a triangle is equals to the sum of measures of its interior opposite angles.

In order words:

Exterior Angle = Sum of Interior Opposite Angles

As shown in the following diagram:


Exterior Angle is ∠ ACD and its two interior opposite angles are ∠ BAC and ∠ ABC
Measure of ∠ ACD = 130°
Measure of one interior ∠ BAC = 60°
Measure of other interior ∠ ABC = 70°

Now, we can observe that:
130° = 60° + 70°
Or we can say:
∠ ACD = ∠ BAC and ∠ ABC

Hence, its demonstrated that measure of exterior angle of a triangle is equals to the sum of measures of its interior opposite angles



Now let's prove/justify External Angle Property of a Triangle:

Before we prove/justify this topic, you are advised to read:

What is Angle Sum Property of a Triangle ?
What is Linear Pair ?

External Angle Property of a Triangle is proved or justified as follows:

In the following diagram:


∠ 1 = External Angle
∠ 2, ∠ 3 & ∠ 4 = Interior Angles or we can also refer them as Angles of Triangle ABC
∠ 3 & ∠ 4 = Interior Opposite Angles to Exterior ∠ 1

Now, as per Angle Sum Property:
∠ 2 + ∠ 3 + ∠ 4 = 180°
Or
∠ 2 = 180° - ∠ 3 - ∠ 4 ..... (Statement 1)

And, we can see that ∠ 1 & ∠ 2 form a Linear Pair, therefore:
∠ 1 + ∠ 2 = 180°
Or
∠ 1 = 180° - ∠ 2

Put values of ∠ 2 from statement 1 and we get:
∠ 1 = 180° - (180 - ∠ 3 - ∠ 4)

Open brackets and we get:
∠ 1 = 180 - 180 + ∠ 3 + ∠ 4

Solve R.H.S and we get
∠ 1 = ∠ 3 + ∠ 4 ..... (statement 2)

Now as mentioned above:
∠ 1 = External Angle
∠ 3 & ∠ 4 = Interior Opposite Angles to Exterior ∠ 1
Put these into statement 2 and we get:
Exterior ∠ 1 = Interior Opposite ∠ 3 + Interior Opposite ∠ 4

Hence, proved:
Exterior = Sum of Interior Opposite Angles

From this property we can also conclude that:
Exterior Angle of a Triangle is Greater Than either of its Interior Opposite Angles.

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Study More Solved Questions / Examples

  • In the following Triangle ABC, find measure of exterior ∠ 1:

  • Find measure of exterior angle whose interior opposite angles are 50° & 75°
  • Find error in the following diagram:


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