What theorem of triangle inequality states that the measure of an exterior angle of a triangle is greater that the measure of either of the two nonadjacent interior angles?

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What is the Exterior Angle of a Triangle?
Triangle Exterior Angle Theorem
How to Find the Exterior Angles of a Triangle?

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ACTIVITY 1 Determine the common ratio and solve the next two terms of the following geometric sequence. 1.80, 20,52.45,30,20 3. 4x,16x4,64x7

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III. Valuing. The mathematical skills we apply in dealing with sequences and series can be translated to real life-skills necessary for our daily surv … ival. Consciously or unconsciously, our social and emotional skills are sharpened when we go through every learning experience. Given the identified mathematical skill, think of a parallel real-life skill that can also be developed while solving problems on sequences and series. Use number one as an example. 1. Order of the terms in a sequence - In setting our priorities, we should learn how to weigh things and finish the tasks that are of great importance (first things first), then the rest follows. In this way, we can prevent unnecessary stress due to unfinished tasks or late submissions.2. Identifying the givens in a sequence/series problem 3. Trial and error in finding the missing terms

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The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles of the triangle.

m ∠ 4 = m ∠ 1 + m ∠ 2

Proof:

Given: Δ P Q R

To Prove: m ∠ 4 = m ∠ 1 + m ∠ 2

	Statement	Reason
1	Δ P Q R is a triangle	Given
2	m ∠ 1 + m ∠ 2 + m ∠ 3 = 180 °	Triangle Sum Theorem
3	∠ 3 and ∠ 4 form a linear pair	Definition of linear pair.
4	∠ 3 and ∠ 4 are supplementary	If two angles form a linear pair, they are supplementary.
5	m ∠ 3 + m ∠ 4 = 180 °	Definition of supplementary angles.
6	m ∠ 3 + m ∠ 4 = m ∠ 1 + m ∠ 2 + m ∠ 3	Statements 2, 5 and Substitution Property.
7	m ∠ 4 = m ∠ 1 + m ∠ 2	Subtraction Property.

So, we all know that a triangle is a 3-sided figure with three interior angles. But there exist other angles outside the triangle, which we call exterior angles.

We know that the sum of all three interior angles is always equal to 180 degrees in a triangle.

Similarly, this property holds for exterior angles as well. Also, each interior angle of a triangle is more than zero degrees but less than 180 degrees. The same goes for exterior angles.

In this article, we will learn about:

Triangle exterior angle theorem,
exterior angles of a triangle, and,
how to find the unknown exterior angle of a triangle.

What is the Exterior Angle of a Triangle?

The exterior angle of a triangle is the angle formed between one side of a triangle and the extension of its adjacent side.

In the illustration above, the triangle ABC’s interior angles are a, b, c, and the exterior angles are d, e, and f. Adjacent interior and exterior angles are supplementary angles.

In other words, the sum of each interior angle and its adjacent exterior angle is equal to 180 degrees (straight line).

Triangle Exterior Angle Theorem

The exterior angle theorem states that the measure of each exterior angle of a triangle is equal to the sum of the opposite and non-adjacent interior angles.

Remember that the two non-adjacent interior angles opposite the exterior angle are sometimes referred to as remote interior angles.

For example, in triangle ABC above;

⇒ d = b + a

⇒ e = a + c

⇒ f = b + c

Properties of exterior angles

An exterior angle of a triangle is equal to the sum of the two opposite interior angles.
The sum of exterior angle and interior angle is equal to 180 degrees.

⇒ c + d = 180°

⇒ a + f = 180°

⇒ b + e = 180°

All exterior angles of a triangle add up to 360°.

Proof:

⇒ d + e + f = b + a + a + c + b + c

⇒ d +e + f = 2a + 2b + 2c

= 2(a + b + c)

But, according to triangle angle sum theorem,

a + b + c = 180 degrees

Therefore, ⇒ d +e + f = 2(180°)

= 360°

How to Find the Exterior Angles of a Triangle?

Rules to find the exterior angles of a triangle are pretty similar to the rules to find the interior angles. It is because wherever there is an exterior angle, there is an interior angle with it, and both add up to 180 degrees.

Let’s take a look at a few example problems.

Example 1

Given that for a triangle, the two interior angles 25° and (x + 15) ° are non-adjacent to an exterior angle (3x – 10) °, find the value of x.

Solution

Apply the triangle exterior angle theorem:

⇒ (3x − 10) = (25) + (x + 15)

⇒ (3x − 10) = (25) + (x +15)

⇒ 3x −10 = x + 40

⇒ 3x – 10 = x + 40

⇒ 3x = x + 50

⇒ 2x = 50

x =25