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Document related concepts
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Transcript 
ANGLES
KHM
Polygons
Definition: A closed figure formed by a finite number of coplanar
segments so that each segment intersects exactly two
others, but only at their endpoints.
These figures are not polygons
These figures are polygons
Complementary vs. Supplementary
Angles
Complementary Angles: sum of the angles is 90 degrees
Supplementary Angles: sum of the angels is 180 degrees
Interior Angle of a Polygon
The interior angles of a polygon are the angles
inside the polygon, formed by two adjacent sides.
For example, ∆ABC has interior
angles:
ABC,
BAC,
BCA
Exterior Angle of a Polygon
An exterior angle of a polygon is an angle that
forms a linear pair with an interior angle. It is an
angle outside the polygon formed by one side and
one extended side of the polygon.
For example, ∆ABC has
exterior angle:
Exterior Angle
Interior Angles
ACD. It forms a linear
pair with ACB.
A
D
B
C
What is the sum of the measures of the interior
angles of a convex n-gon?
Polygon
Number of
Sides
Sum of Measures
of Interior Angles
Triangle
3
180°
Quadrilateral
4
360°
Pentagon
5
540°
Hexagon
6
720°
n-gon
n
(n - 2)180°
Angles in a triangle
c
a
b
For any triangle,
a + b + c = 180°
The angles in a triangle add up to 180°.
Calculating angles in a triangle
Calculate the size of the missing angles in each of the following triangles.
116°
33°
a
31°
64°
b
326°
82°
49°
43°
25°
d
88°
c
28°
233°
Exterior Angle Theorem
a + b + c = 180
b
c + d = 180
a
c
d
so….. a + b = d
Theorem: An measure of an exterior angle of a triangle
is equal to the sum of the measures of the two nonadjacent interior angles.
More Examples
Calculating angles
Calculate the size of the lettered angles in each of the following triangles.
116°
b
33°
a
64°
82°
31°
34°
43°
c
25°
d
131°
152°
127°
272°
Angles made with parallel lines
When a straight line crosses two parallel lines eight angles are formed.
a
b
d
c
e
f
h
g
Which angles are equal to each other?
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
d = h because
Corresponding angles are equal
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
a = e because
Corresponding angles are equal
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
c = g because
Corresponding angles are equal
Corresponding angles
There are four pairs of corresponding angles, or F-angles.
a
b
d
c
e
f
h
g
b = f because
Corresponding angles are equal
Alternate angles
There are two pairs of alternate angles, or Z-angles.
a
b
d
c
e
f
h
g
d = f because
Alternate angles are equal
Alternate angles
There are two pairs of alternate angles, or Z-angles.
a
b
d
c
e
f
h
g
c = e because
Alternate angles are equal
Interior and exterior angles in a triangle
Any exterior angle in a triangle is equal to the sum of the two
opposite interior angles.
c
ca
b
b
a=b+c
We can prove this by constructing a line parallel to this side.
These alternate angles are equal.
These corresponding angles are equal.
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