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Polygon Properties
Recall: Definition of Polygons
• Side lengths• Angle measures• Types
POLYGON ANGLE SUM
3 sides 1 triangle
4 sides
5 sides
6 sides
7 sides
8 sides
POLYGON ANGLE SUM
3 sides 1 triangle
4 sides
5 sides
6 sides
7 sides
8 sides
POLYGON ANGLE SUM
3 sides 1 triangle
4 sides 2 triangles
5 sides 3 triangles
6 sides 4 triangles
7 sides 5 triangles
8 sides 6 triangles
POLYGON ANGLE SUM
3 sides 1 triangle 1(180)=180
4 sides 2 triangles 2(180)=360
5 sides 3 triangles 3(180)=540
6 sides 4 triangles 4(180)=720
7 sides 5 triangles 5(180)=900
8 sides 6 triangles 6(180)=1080
POLYGON SUM
n-gon n sides (n-2) triangles
(n-2)180
POLYGON ANGLE SUM THEOREM
The sum of the measures of the n angles of an n-gon is (n-2)180.
Recall: Characteristics of Quads
• Kite• Trapezoid
– Isosceles• Parallelogram
– Rhombus– Rectangle– Square
Trapezoids
A trapezoid is a quadrilateral with exactly one pair of parallel sides.
A B
CD
O
M N
P
The parallel sides are called bases. The non-parallel sides are called legs.
When the legs of the trapezoid are congruent, it is called an isosceles trapezoid.
The segment joining the midpoints of the legs is called the midsegment or median.
Trapezoid-Midsegment Conjecture
A
B C
D
E
M NThe midsegment of the trapezoid is parallel to the bases, and has a length equal to the average of the lengths of the bases.
Isosceles Trapezoid:RECALL: An isosceles trapezoid is a trapezoid with congruent legs.Each pair of base angles of an isosceles trapezoid are congruent.The diagonals of an isosceles trapezoid are congruent.
A
B C
D /_A is congruent to /_D
/_B is congruent to /_C
Segment AC segment BD
Example
A
B C
D
X Y
Suppose ABCD is isosceles with AB = CD
IF AX = 4, what is CD?IF mABC = 110, what is mBAD?IF mBAD = 65, what is mCDA?IF AD = 22, BC = 10, what is XY?IF BC = 20, XY = 32, what is AD?
Kites
A KITE is a quadrilateral with exactly two pairs of congruent sides.The angles included by the congruent sides are called VERTEX ANGLES. The two other angles are called NON-VERTEX ANGLES. Segment AC and segment BD are the DIAGONALS
A
B
C
D
PROPERTIES OF KITESKite Angles Conjecture: The non-vertex angles of a kite are congruent.
B D
A
B
CD
PROPERTIES OF KITESKite Angle Bisector Conjecture: The vertex angles of a kite are bisected by a diagonal.
AC bisects BAD and BCD.
A
B
CD
PROPERTIES OF KITESKite Diagonals Conjecture: The diagonals of a kite are perpendicular.
Segment AC segment BD
A
B
CD
PROPERTIES OF KITESKite Diagonal Bisector Theorem: The diagonal connecting the vertex angles of a kite is the perpendicular bisector of the other diagonal.
Segment AC is the bisector of segment BD
A
B
CD
