9-2 Polygons

More About Polygons Congruent Segments and Angles Regular Polygons Triangles and Quadrilaterals Hierarchy Among Polygons

Simple curve – a curve that does not cross itself, except that if you draw it with a pencil, the starting and stopping points may be the same.

Closed curve – a curve that can be started and stopped at the same point.

Polygon – a simple, closed curve with sides that are line segments.

Convex curve – a simple, closed curve with no indentations; the segment connecting any two points in the interior of the curve is wholly contained in the interior of the curve.

Concave curve – a simple, closed curve that is not convex; it has an indentation.

Curves and Polygons

NOW TRY THIS 9-6 Page 590

Draw a curve that is neither simple nor closed.

This curve crosses (not simple) and is open on the ends

(not closed)

Polygonal Regions

Polygons and their interiors together are called polygonal regions

Is point X inside or outside the simple closed curve?

Answer: Outside, one approach is to start shading the area surrounding point X. If we stay between the lines, we should be able to decide if the shaded area is inside or outside the curve.

NOW TRY THIS 9-7 Page 591

More About PolygonsPolygons are classified according to the number of sides or vertices they have.

More About Polygons (cont.)Interior angle or Angle of the polygon – any angle formed by two

sides of a polygon with a common vertex.

Exterior angle of a convex polygon – any angle formed by a side of a polygon and the extension of a continuous side of the polygon.

Diagonal – a line segment connecting nonconsecutive vertices of a polygon.

Congruent Segments and AnglesCongruent parts – are parts that are the same size and shape.

Two line segments are congruent if a tracing of one line segment can be fitted exactly on top of the other line segment.

Two angles are congruent if they have the same measure.

Regular PolygonsAll sides are congruent and all angles are congruent.

A regular polygon is equilateral and equiangular.

A regular triangle is an equilateral triangle.

EXAMPLE: The Pentagon

EXAMPLE: Hexnut

Triangles and Quadrilaterals

Right triangle one right angle

Acute triangle all angles are acute

Obtuse triangle one obtuse angle

Scalene triangle has no congruent sides

Isosceles triangle has at least two congruent sides

Equilateral triangle three congruent sides

Triangles may be classified according to their angle measures. Quadrilaterals are shown below (page 593)

Trapezoid - at least one pair of parallel

sides

Kite – two adjacent sides congruent other two also congruent

Isosceles trapezoid – exactly one pair of congruent sides

Parallelogram – each pair of opposite sides

is parallel

Rectangle – has a right angle

Rhombus – two adjacent sides

congruent

Square - two adjacent sides

congruent

Triangles and Quadrilaterals

Every equilateral triangle is isosceles triangle.

Every triangle is a polygon.

Every isosceles is not equilateral.

The set of all triangles is a proper subset of the set of all polygons.

The set of all equilateral triangles is a proper subset of all isosceles triangles

Hierarchy Among Polygons

NOW TRY THIS 9-9 Page 597Use the definition in Table 9-6 to experiment with several drawings to decide which of the following are true.

1) An equilateral triangle is isosceles. TRUE2) A square is a regular equilateral. TRUE3) If one angle of a rhombus is a right angle, then all

angles of the rhombus are right angles. TRUE4) A square is a rhombus with a right angle. TRUE5) All the angles of a rectangle are right angles. TRUE6) A rectangle is an isosceles trapezoid. TRUE7) Some isosceles trapezoids are kites. TRUE8) If a kite has a right angle, then it must be a square. FALSE

Example #2 Page 597What is the maximum number of intersection points between a quadrilateral and a triangle (where no sides of the polygons are on the same line)?

Answer:A segment can pass through at most two sides of a triangle. If each side of the quadrilateral passes through two sides of the triangle there will then be eight intersections.

Example:

Example #4 Page 597Which of the following figures are convex and which are concave? Why?

Convex Concave

Convex

Concave

Concave – It is possible to connect two points of the figure with a segment that lies partially or fully inside the figure.

Convex – A segment connecting any two points would lie fully inside the figure.

A. B.

C.

D.

Example #6 Page 598Determine how many diagonals each of the following has. Justify your answer.

a) Decagon: has 10 side, the number of ways that all the vertices in a decagon can be connected two at a time is the number of combinations of 10 vertices chosen two at a time, 10C2 = 45. This number includes both diagonals and sides. Subtracting the number of sides, 10 from 45 yields 35 diagonals in a decagon.

b) 20-gon: has 20 side, the number of ways that all the vertices in a 20-gon can be connected two at a time is the number of combinations of 20 vertices chosen two at a time, 20C2 = 190. This number includes both diagonals and sides. Subtracting the number of sides, 20 from 190 yields 170 diagonals in a 20-gon.

Example #6 Page 598Determine how many diagonals each of the following has. Justify your answer.

2( 1) ( 1) 2 ( 3) yields

2 2 2 2

n n n n n n n n nn

c) 100-gon: has 100 side, the number of ways that all the vertices in a decagon can be connected two at a time is the number of combinations of 100 vertices chosen two at a time, 100C2 = 4950. This number includes both diagonals and sides. Subtracting the number of sides, 100 from 4950 yields 4850 diagonals in a 100-gon.

d) N-gon - has n side, the number of ways that all the vertices in a decagon can be connected two at a time is the number of combinations of n vertices chosen two at a time.

This number includes both diagonals and sides. Subtracting the number of sides, n from

2

! ( 1)

( 2)!2! 2n

n n nC

n

Example #8 Page 598Describe the regions (a) and (b) in the following Venn Diagram, where the universal set is the set of all parallelograms.

Parallelograms

(a) (b)Squares

(a) and (b) represent rhombuses and rectangles. A square is a parallelogram with all sides congruent (rhombus) and all right angles (rectangle). If each circle represents one of these properties, the intersection contains a parallelogram with both, or a square.

HOMEWORK 9-2

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