curve—continuous set of points (includes lines)curve—continuous set of points (includes lines)

simple curves—do not intersect themselvessimple curves—do not intersect themselves

closed curves—start and end at same pointclosed curves—start and end at same point

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

Types of PolygonsTypes of Polygons

convexconvexconcaveconcave

side—segment making up part of a polygonside—segment making up part of a polygon

vertex—corner of a polygon; where two sides meetvertex—corner of a polygon; where two sides meet

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.

acute triangle: three acute anglesacute triangle: three acute angles

60°60°

75°75°45°45°

right triangle: one right angleright triangle: one right angle

50°50°

40°40°

obtuse triangle: one obtuse angleobtuse triangle: one obtuse angle

20°20°

30°30° 130°130°

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

BB

AA CC

ABC is obtuse scalene. ABC is obtuse scalene.

EE

DD FF

DEF is right isosceles. DEF is right isosceles.

HH

GG II

GHI is acute equilateral. GHI is acute equilateral.

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

m A + 80 + 55 + 120 = 360m A + 80 + 55 + 120 = 360m A + 255 = 360m A + 255 = 360

m A = 105°m A = 105°m A + 255 – 255 = 360 – 255m A + 255 – 255 = 360 – 255

Example 5Example 5