Exterior Angles In A Triangle - MathBitsNotebook (Geo - CCSS Math)

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Exterior Angles in a Triangle MathBitsNotebook.com Topical Outline | Geometry Outline | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts divider

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
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. Solution:∠BDC is an exterior angle for ΔABD. m∠BDC = 35 + 25 m∠BDC = 60º 180 = m∠DBC + 60 + 60 m∠DBC = 60º 		4. exWhite2 Find xº. Solution: 100 = x + 50 x = 50º

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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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Topical Outline | Geometry Outline | MathBitsNotebook.com | MathBits' Teacher Resources Terms of Use Contact Person: Donna Roberts
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