 § 10.1 Naming Polygons Naming PolygonsNaming Polygons  § 10.4 Areas of Triangles and Trapezoids  § 10.3 Areas of Polygons Areas of PolygonsAreas of Polygons  § 10.2 Diagonals and Angle Measure Diagonals and Angle MeasureDiagonals and Angle Measure  § 10.6 Symmetry  § 10.5 Areas of Regular Polygons  § 10.7 Tessellations

You will learn to name polygons according to the number of _____ and ______. 1) regular polygon 2) convex 3) concave sidesangles

A polygon is a _____________ in a plane formed by segments, called sides. closed figure A polygon is named by the number of its _____ or ______. sidesangles A triangle is a polygon with three sides. The prefix ___ means three. tri

Prefixes are also used to name other polygons. PrefixNumber of Sides Name of Polygon tri- quadri- penta- hexa- hepta- octa- nona- deca triangle quadrilateral pentagon hexagon heptagon octagon nonagon decagon

U T S Q R P A vertex is the point of intersection of two sides. A segment whose endpoints are nonconsecutive vertices is a diagonal. Consecutive vertices are the two endpoints of any side. Sides that share a vertex are called consecutive sides.

An equilateral polygon has all _____ congruent. An equiangular polygon has all ______ congruent. sides angles A regular polygon is both ___________ and ___________. equilateral equiangular equilateral but not equiangular but not equilateral regular, both equilateral and equiangular Investigation: As the number of sides of a series of regular polygons increases, what do yousides of a series of regular polygons increases notice about the shape of the polygons?

A polygon can also be classified as convex or concave. If all of the diagonals lie in the interior of the figure, then the polygon is ______. convex If any part of a diagonal lies outside of the figure, then the polygon is _______. concave

You will learn to find measures of interior and exterior angles of polygons. Nothing New!

Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral (180) = 360 1) Draw a convex quadrilateral. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? Make a table like the one below.

Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral (180) = 360 1) Draw a convex pentagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? pentagon (180) = 540

Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral (180) = 360 1) Draw a convex hexagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? pentagon (180) = 540 hexagon (180) = 720

Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral (180) = 360 1) Draw a convex heptagon. 2) Choose one vertex and draw all possible diagonals from that vertex. 3) How many triangles are formed? pentagon (180) = 540 hexagon (180) = 720 heptagon (180) = 900

Convex Polygon Number of Sides Number of Diagonals from One Vertex Number of Triangles Sum of Interior Angles quadrilateral (180) = 360 1) Any convex polygon. 2) All possible diagonals from one vertex. 3) How many triangles? pentagon (180) = 540 hexagon (180) = 720 heptagon (180) = 900 n -gon n n - 3 n - 2 ( n – 2)180 Theorem 10-1 If a convex polygon has n sides, then the sum of the measure of its interior angles is (n – 2)180.

57° 48° 74° 55° 54° 72° In §7.2 we identified exterior angles of triangles. Likewise, you can extend the sides of any convex polygon to form exterior angles. The figure suggests a method for finding the sum of the measures of the exterior angles of a convex polygon. When you extend n sides of a polygon, n linear pairs of angles are formed. The sum of the angle measures in each linear pair is 180. sum of measure of exterior angles sum of measures of linear pairs sum of measures of interior angles = = – –n180180(n – 2) =–180n180n =360 sum of measure of exterior angles

Theorem 10-2 In any convex polygon, the sum of the measures of the exterior angles, (one at each vertex), is 360. Java Applet

You will learn to calculate and estimate the areas of polygons. 1) polygonal region 2) composite figure 3) irregular figure

Any polygon and its interior are called a ______________. polygonal region In lesson 1-6, you found the areas of rectangles. Postulate 10-1 Area Postulate For any polygon and a given unit of measure, there is a unique number A called the measure of the area of the polygon. Area can be used to describe, compare, and contrast polygons. The two polygons below are congruent. How do the areas of these polygons compare? They are the same. Postulate 10-2 Congruent polygons have equal areas.

The figures above are examples of ________________. composite figures They are each made from a rectangle and a triangle that have been placed together. You can use what you know about the pieces to gain information about the figure made from them. You can find the area of any polygon by dividing the original region into smaller and simpler polygon regions, like _______, __________, and ________. rectangles squares triangles The area of the original polygonal region can then be found by __________ _________________________. adding the areas of the smaller polygons

Postulate 10-3 Area Addition Postulate The area of a given polygon equals the sum of the areas of the non-overlapping polygons that form the given polygon. Area Total = A 1 + A 2 + A3A

3 units Area of Square 3u X 3u = 9u 2 Area of Rectangle 1u X 2u = 2u 2 Area of Rectangle Area of Square Find the area of the polygon in square units. Area of polygon = = 7u 2

You will learn to find the areas of triangles and trapezoids. Nothing new!

b h Look at the rectangle below. Its area is bh square units. The diagonal divides the rectangle into two _________________. congruent triangles The area of each triangle is half the area of the rectangle, or This result is true of all triangles and is formally stated in Theorem 10-3.

Consider the area of this rectangle A (rectangle) = bh Base Height

Theorem 10-3 Area of a Triangle If a triangle has an area of A square units, b h a base of b units, and a corresponding altitude of h units, then

Find the area of each triangle: A = 13 yd 2 6 yd 18 mi 23 mi A = 207 mi 2

Because the opposite sides of a parallelogram have the same length, the area of a parallelogram is closely related to the area of a ________. rectangle The area of a parallelogram is found by multiplying the ____ and the ______. base height base height Base – the bottom of a geometric figure. Height – measured from top to bottom, perpendicular to the base. Next we will look at the area of trapezoids. However, it is helpful to first understand parallelograms.

h b1b1 b2b2 b1b1 b2b2 Starting with a single trapezoid. The height is labeled h, and the bases are labeled b 1 and b 2 Construct a congruent trapezoid and arrange it so that a pair of congruent legs are adjacent. The new, composite figure is a parallelogram. It’s base is ( b 1 + b 2 ) and it’s height is the same as the original trapezoid. The area of the parallelogram is calculated by multiplying the base X height. A (parallelogram) = h(b 1 + b 2 ) The area of the trapezoid is one-half of the parallelogram’s area.

Theorem 10-4 Area of a Trapezoid If a trapezoid has an area of A square units, h b1b1 b2b2 bases of b 1 and b 2 units, and an altitude of h units, then

Find the area of the trapezoid: A = 522 in 2 18 in 20 in 38 in

You will learn to find the areas of regular polygons. 1) center 2) apothem

Every regular polygon has a ______, a point in the interior that is equidistant from all the vertices. center A segment drawn from the center that is perpendicular to a side of the regular polygon is called an ________. apothem In any regular polygon, all apothems are _________. congruent

72° s a The figure below shows a center and all vertices of a regular pentagon. There are 5 vertices and each is 72° from the other (360 ÷ 5 = ___.) 72 An apothem is drawn from the center, and is _____________ to a side. perpendicular Now, create a triangle by drawing segments from the center to each vertex on either side of the apothem. The area of a triangle is calculated with the following formula: Now multiply this times the number of triangles that make up the regular polygon. What measure does 5s represent? perimeter Rewrite the formula for the area of a pentagon using P for perimeter.

Theorem 10-5 Area of a Regular Polygon If a regular polygon has an area of A square units, an apothem of a units, and a perimeter of P units, then P

5.5 ft 8 ft Find the area of the shaded region in the regular polygon. Area of polygon Area of triangle To find the area of the shaded region, subtract the area of the _______ from the area of the ________: triangle pentagon The area of the shaded region: 110 ft 2 – 22 ft 2 = 88 ft 2

Find the area of the shaded region in the regular polygon. Area of polygon Area of triangle To find the area of the shaded region, subtract the area of the _______ from the area of the ________: triangle hexagon The area of the shaded region: m 2 – 55.2 m 2 = m m 8 m

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