definition
An exterior angle of a triangle is an angle formed by one side of the triangle and the extension of an adjacent side of the triangle.

    exdiagram1
FACTS:
• Every triangle has 6 exterior angles, two at each vertex.
• Angles 1 through 6 are exterior angles.
• Notice that the "outside" angles that are "vertical" to the angles inside the triangle are
NOT called exterior angles of a triangle.


theorem1
The measure of an exterior angle of a triangle is equal to the sum of the measures of the two non-adjacent interior angles.
(Non-adjacent interior angles may also be referred to as remote interior angles.)

extdiagram

exdiagram2
FACTS:
• An exterior ∠ is equal to the addition of the two Δ angles not right next to it.
140º = 60º + 80º;        120º = 80º + 40º;
100º = 60º + 40º
• An exterior angle is supplementary to its adjacent Δ angle.
140º is supp to 40º
• The 2 exterior angles at each vertex are = in measure because they are vertical angles.
• The exterior angles (taken one at a vertex) always total 360º

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So, how do we know that this theorem is true?

This theorem is connected to the theorem that states "the sum of the measures of the angles of a triangle = 180º ", and the concept that a straight line (angle) = 180º.

Let's take a look:
extdiagram1a extdiagramab

If we pull the two equation statements together, we can see the connection:

extdiagram4

The 140º can replace the 80º + 60º.
In other words, the exterior angle's measure is the same as the measures of the two non=adjacent interior angles added together.

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There is a "common sense" inequality theorem about exterior angles:

theorem1
The measure of the exterior angle of a triangle is greater than the measure of either non-adjacent interior angle.

In the diagram at the right,
∠1 is an exterior angle for ΔABC.
Since, by the previous theorem, m∠1 = m∠2 + m∠3,
it is common sense that m∠1 > m∠ 2
and m∠1 > m∠3.

ineq6

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Examples:

1. exYellow
Solution: Using the Exterior Angle Theorem
145 = 80 + x
x = 65

Now, if you forget the Exterior Angle Theorem, you can still get the answer by noticing that a straight angle has been formed at the vertex of the 145º angle. See Example 2.
2. exDiagram2
Solution: "I forgot the Exterior Angle Theorem."
The angle adjacent to 145º will form a straight angle along with 145º adding to 180º. That angle is 35º.
Now use rule that sum of ∠s in Δ = 180º.
35 + 80 + x = 180
115 + x = 180
x = 65
3. exWhite

Find m∠DBC.
When a diagram contains more than one triangle, an exterior angle can exist as an interior angle from another triangle.

Solution:∠BDC is an exterior angle for ΔABD.
m∠BDC = 35 + 25
m∠BDC = 60º
180 = m∠DBC + 60 + 60
m∠DBC =
60º

Alternative solution using the fact that the measures of the angles in ΔABC add to 180º.
mA + mC + mDBC + mDBA = 180
º
35º + 60º + mDBC + 25º = 180º
120º + mDBC = 180º
mDBC = 60º

4. exWhite2
Find xº.
Use the fact that the 100º∠ can be an exterior angle for ΔADB.

Solution:
100 = x + 50
x = 50º

Alternative solution using the fact that the measures of the angles in ΔABC and ΔDBC each add to 180º.
Find mC. 30º + 100º + mC = 180. mC = 50º
Now, use mA + mC + mDBC + mDBA = 180
º
50º + 50º + 30º + xº = 180º
130º + x = 180º
x = 50º
Alternate solution using a linear pair: Find that m∠ADB = 80º. The use the sum of the angles ΔADB to find x.

5.
ineq7

Solution:
1) Exterior Angle Theorem - TRUE

2) Inequality Theorem about Exterior Angles (stated above) - TRUE

3)
Linear Pairs are supplementary (2 ∠s adding to 180) - TRUE


4)
FALSE (it should read m∠1 > m∠C)

 
Given ΔABC as shown.
Which statement is NOT true?
1) m∠1 = m∠A + m∠C
2) m∠1 > m∠A
3) m∠1 + m∠ABC = 180º
4) m∠1 < m∠C



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