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simpleclosedsimpleclosed
simple,not closed
simple,not closed
simpleclosedsimpleclosed
closed, not simple
closed, not simple
A polygon is a simple closed plane figure made up of line segments.
A polygon is a simple closed plane figure made up of line segments.
PolygonPolygon
A convex polygon is a polygon in which all interior angles have measures less than 180°.
A convex polygon is a polygon in which all interior angles have measures less than 180°.
Convex PolygonConvex Polygon
A concave polygon is a polygon in which at least one interior angle has a measure greater than 180°.
A concave polygon is a polygon in which at least one interior angle has a measure greater than 180°.
Concave PolygonConcave Polygon
Number of SidesNumber of Sides NameName
quadrilateralquadrilateral
triangletriangle
pentagonpentagon
hexagonhexagon
heptagonheptagon
33
44
55
66
77
Number of SidesNumber of Sides NameName
nonagonnonagon
octagonoctagon
decagondecagon
dodecagondodecagon
n-gonn-gon
88
99
1010
1212
nn
regular polygon—all sides same length, all angles same measure
regular polygon—all sides same length, all angles same measure
Example 1Example 1Name each polygon and indicate whether it is regular.Name each polygon and indicate whether it is regular.
ABCD is a quadrilateral.ABCD is a quadrilateral.
DD
AA BB
CC
Name each polygon and indicate whether it is regular.Name each polygon and indicate whether it is regular.
EFGHIJ is a regular hexagon.EFGHIJ is a regular hexagon.
JJ
EE
GG
HHII
FF
Example 1Example 1
scalene triangle: A triangle with no two sides equal in length.
scalene triangle: A triangle with no two sides equal in length.
isosceles triangle: A triangle with at least two sides equal in length.
isosceles triangle: A triangle with at least two sides equal in length.
equilateral triangle: A triangle with all sides equal in length.equilateral triangle: A triangle with all sides equal in length.
Classify each triangle according to the measures of its angles and the lengths of its sides.
Classify each triangle according to the measures of its angles and the lengths of its sides.
Example 2Example 2
TheoremTheoremIn any triangle, the sum of the measures of the angles is 180°: in ABC m A + m B + m C = 180°.
In any triangle, the sum of the measures of the angles is 180°: in ABC m A + m B + m C = 180°.
Write and solve an equation to find m X.Write and solve an equation to find m X.
XX
YY ZZ
??
116°116° 22°22°
Example 3Example 3
m X + m Y + m Z = 180m X + m Y + m Z = 180m X + 116 + 22 = 180m X + 116 + 22 = 180
m X + 138 = 180m X + 138 = 180m X = 42°m X = 42°
Example 3Example 3
A quadrilateral is a four-sided polygon.A quadrilateral is a four-sided polygon.
QuadrilateralQuadrilateral
A trapezoid is a quadrilateral with at least one pair of parallel sides.
A trapezoid is a quadrilateral with at least one pair of parallel sides.
TrapezoidTrapezoid
A parallelogram is a quadrilateral with two pairs of parallel sides.
A parallelogram is a quadrilateral with two pairs of parallel sides.
ParallelogramParallelogram
A rectangle is a parallelogram with four right angles.
A rectangle is a parallelogram with four right angles.
RectangleRectangle
A rhombus is a parallelogram with four congruent sides.
A rhombus is a parallelogram with four congruent sides.
RhombusRhombus
A square is a rectangle with four congruent sides.A square is a rectangle with four congruent sides.
SquareSquare
A diagonal is a line segment joining any two nonadjacent vertices of a polygon.
A diagonal is a line segment joining any two nonadjacent vertices of a polygon.
DiagonalDiagonal
number of diagonals from one vertex
number of diagonals from one vertexnumber of sides, nnumber of sides, nnumber of triangles formednumber of triangles formed
TriangleTriangle
sum of angle measuressum of angle measures
33
11
180°180°
number of diagonals from one vertex
number of diagonals from one vertexnumber of sides, nnumber of sides, nnumber of triangles formednumber of triangles formed
QuadrilateralQuadrilateral
sum of angle measuressum of angle measures
44
22
2(180) = 360°2(180) = 360°
number of diagonals from one vertex
number of diagonals from one vertexnumber of sides, nnumber of sides, nnumber of triangles formednumber of triangles formed
PentagonPentagon
sum of angle measuressum of angle measures
55
33
3(180) = 540°3(180) = 540°
number of diagonals from one vertex
number of diagonals from one vertexnumber of sides, nnumber of sides, nnumber of triangles formednumber of triangles formed
HexagonHexagon
sum of angle measuressum of angle measures
66
44
4(180) = 720°4(180) = 720°
Find the sum of the measures of the angles in a regular octagon.
Find the sum of the measures of the angles in a regular octagon.
6(180)6(180) = 1,080°= 1,080°
Example 4Example 4
Find the measure of each angle in a regular octagon.Find the measure of each angle in a regular octagon.
= 135°= 135°1,0808
1,0808
Example 4Example 4
Write and solve an equation to find m A. Write and solve an equation to find m A.
BB
AA
CC
120°120°
55°55°
DD
80°80°
Example 5Example 5