Download 7.4 Sum of Exterior Angles

Bellwork
• If the measure of each interior angle
of a regular polygon is 172° find the
number of sides of a polygon.
• An angle of a triangle is 50. Find the
measure of the angle formed by the
angle bisectors of the other two
angles. (Hint:Draw a picture)
Exterior Angles (defn)
• Name the exterior angle of this
triangle?
Sum of the
Exterior Angles
7.4
Exterior Angle Summary
Se=360
E=
360
n
• How many exterior angles are in a
heptagon?
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# of Diagonals Formula
A proof written out!
Next I will prove algebraically that the sum of the exterior
angles of a convex polygon is 360°. (Just so you know why!)
•d=
n(n − 3)
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1. We know the sum of straight angles at the vertices of a
convex n-gon is n(180°).
2. We know the sum of the interior angles of a convex n-gon
is 180°(n-2).
3. Therefore the sum of the exterior angles of a convex
polygon could be represented by:
n(180°) - 180°(n-2).
4. Factoring the expression, we get: 180°[n – (n-2)].
5. Simplifying, we get 180°(n – n + 2) = 180°(2) = 360°.
Thus, the sum of the exterior angles of a convex polygon is
360°. **This is the important thing to remember!**
Example 1
• Find the sum of the exterior angles
of a convex heptagon.
Example 2
• If the measure of an exterior angle
of a regular polygon is 15, how
many sides does the polygon have?
• What is the measure of an exterior
angle of an equiangular heptagon?
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Example 3
• Find the measure of each interior
and exterior angle of a dodecagon.
Example 4
• What is the name of an equiangular
polygon if the ratio of the measure of
an interior angle to the measure of
an exterior angle is 3:1?
Example 5
Example 6
• In what polygon is the sum of the
measures of the exterior angles, one
per vertex, equal to the sum of the
measures of the angles of the
polygon?
• Given: ABCDEF is a regular hexagon
• Prove: ∠BAC ≅ ∠EDF
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Quiz Reminders
• Quiz Tomorrow:
• Know formulas for Sum of Interior, an
Interior, Sum of Exterior, an Exterior, and
diagonals.
• Know what the term “Regular” means, the
names of polygons, Exterior Angle
Theorem and the properties of the
“midline” of a triangle.
• “No Choice Theorem”—Remember that? ☺
Homework
• Pg. 309 # 3, 6, 11b, 13, 14
• Pg. 316 # 1, 3, 5, 6, 7, 11-14
• Quiz tomorrow!
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