

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.



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.


The measure of an exterior angle of a triangle is equal to the sum of the measures of the two nonadjacent interior angles.
(Nonadjacent interior angles may also be referred to as remote interior angles.) 


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

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:
If we pull the two equation statements together, we can see the connection:
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.
There is a "common sense" inequality theorem about exterior angles:

The measure of the exterior angle of a triangle is greater than the measure of either nonadjacent 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.


1. 
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. 
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. 
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.
m∠A + m∠C + m∠DBC + m∠DBA = 180º
35º + 60º + m∠DBC + 25º = 180º
120º + m∠DBC = 180º
m∠DBC = 60º

4.
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
m∠C. 30º + 100º +
m∠C = 180.
m∠C = 50º
Now, use m∠A + m∠C + m∠DBC + m∠DBA = 180º
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.


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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