# CH_9.2_Curves_Polygons_and_Circles

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### Chapter 9: Geometry:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 2 Chapter 9: Geometry 9.1 Points, Lines, Planes, and Angles 9.2 Curves, Polygons, and Circles 9.3 Perimeter, Area, and Circumference 9.4 The Geometry of Triangles: Congruence, Similarity, and the Pythagorean Theorem 9.5 Space Figures, Volume, and Surface Area 9.6 Transformational Geometry 9.7 Non-Euclidean Geometry, Topology, and Networks 9.8 Chaos and Fractal Geometry

### Simple Curve; Closed Curve :

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 6 Simple Curve; Closed Curve A simple curve can be drawn without lifting the pencil from the paper, and without passing through any point twice. A closed curve has its starting and ending points the same, and is also drawn without lifting the pencil from the paper.

### Convex :

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 8 Convex A figure is said to be convex if, for any two points A and B inside the figure, the line segment AB is always completely inside the figure. A B A B Convex Not convex

### Polygons :

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 9 Polygons A polygon is a simple, closed curve made up of only straight line segments. The line segments are called sides , and the points at which the sides meet are called vertices . Polygons with all sides equal and all angles equal are regular polygons .

### Classification of Polygons According to Number of Sides :

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 11 Classification of Polygons According to Number of Sides Number of Sides Name 3 Triangle 4 Quadrilateral 5 Pentagon 6 Hexagon 7 Heptagon 8 Octagon 9 Nonagon 10 Decagon

### Types of Triangles - Angles:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 12 Types of Triangles - Angles All Angles Acute One Right Angle One Obtuse Angle Acute Triangle Right Triangle Obtuse Triangle

### Types of Triangles - Sides:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 13 Types of Triangles - Sides All Sides Equal Two Sides Equal No Sides Equal Equilateral Triangle Isosceles Triangle Scalene Triangle

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 15 Types of Quadrilaterals A square is a rectangle with all sides having equal length. A rhombus is a parallelogram with all sides having equal length.

### Angle Sum of a Triangle:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 16 Angle Sum of a Triangle The sum of the measures of the angles of any triangle is 180 °.

### Example: Finding Angle Measures in a Triangle:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 17 Example: Finding Angle Measures in a Triangle Find the measure of each angle in the triangle below. ( x +20) ° x ° (220 – 3 x ) ° Solution x + x + 20 + 220 – 3 x = 180 – x + 240 = 180 x = 60 Evaluating each expression we find that the angles are 60°, 80° and 40°.

### Exterior Angle Measure:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 18 Exterior Angle Measure The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles. 1 2 3 4 The measure of angle 4 is equal to the sum of the measures of angles 2 and 3 Two other statements can be made.

### Example: Finding Angle Measures in a Triangle:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 19 Example: Finding Angle Measures in a Triangle Find the measure of the exterior indicated below. ( x +20) ° x ° (3 x – 40) ° Solution x + x + 20 = 3 x – 40 2 x + 20 = 3 x – 40 x = 60 Evaluating the expression we find that the exterior angle is 3(60) – 40 =140°.

### Circle:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 20 Circle A circle is a set of points in a plane, each of which is the same distance from a fixed point (called the center ).

### Circle:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 21 Circle A segment with an endpoint at the center and an endpoint on the circle is called a radius (plural: radii ). A segment with endpoints on the circle is called a chord . A segment passing through the center, with endpoints on the circle, is called a diameter . A diameter divides a circle into two equal semicircles . A line that touches a circle in only one point is called a tangent to the circle. A line that intersects a circle in two points is called a secant line .

### Circle:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 22 P R O T Q Circle RT is a tangent line. PQ is a secant line. OQ is a radius. PQ is a chord. O is the center PR is a diameter. PQ is an arc.

### Inscribed Angle:

© 2008 Pearson Addison-Wesley. All rights reserved 9-2- 23 Inscribed Angle Any angle inscribed in a semicircle must be a right angle. To be inscribed in a semicircle, the vertex of the angle must be on the circle with the sides of the angle going through the endpoints of the diameter at the base of the semicircle. 