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.)

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


Examples:
1. 180triangle1
Find m∠B
and m∠C.

Solution: m∠A + m∠B + m∠C = 180
38 + x + (x + 2) = 180
40 + 2x = 180
2x = 140
x = 70 = m∠B
x + 2 = 72 = m∠C

2. The angles in a triangle are represented by (4x - 6)º, (2x + 1)º and (x + 3)º. Is this a right triangle?

Solution:
(4x - 6) + (2x + 1) + (x + 3) = 180
7x - 2 = 180
7x = 182
x = 26
(4x - 6)º = 98º
(2x + 1)º = 53º
(x + 3)º = 29º
No. The triangle is obtuse.

3.180triangle3
m∠ABC=m∠BCD
Find m∠ACD.

Solution: m∠BCD = 56º.
In ΔABC, 85º + 56º + m∠BCA = 180º
m∠BCA = 39º
m∠ACD = 56º - 39º =
17º

4. The angles in a triangle are in the ratio of 1 : 2 : 3. Find the measure of the angles in the triangle.

Solution:
x + 2x + 3x = 180
6x = 180
x = 30
The angles are 30º, 60º, and 90º.

 

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Traditional Proof of the Theorem

The proof of this theorem will utilize linear pairs and the sum of the interior angles of a triangle.
exProofGiven
extproofdiagram
Statements
Reasons
1. exproof2
1. Given
2. ∠2 and ∠4 form a linear pair
2. A linear pair is 2 adjacent ∠s whose non-common sides form opposite rays.
3. ∠2 supp ∠4
3. If 2∠s form a linear pair, they are supplementary.
4.  m∠2 + m∠4 = 180
4. Supplementary ∠s are 2 ∠s the sum of whose measures is 180.
5.  m∠1 + m∠2 + m∠3 = 180
5. The measures of the angles of a triangle add to 180º.
6.  m∠2 + m∠4 = m∠1 + m∠2 + m∠3
6. Substitution
7.  m∠2 = m∠2
7. Reflexive Property (or quantity is = itself)
8.  m∠4 = m∠1 + m∠3
8. Subtraction of Equalities



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