SECTION 9-2 Curves, Polygons, and Circles Slide 9-2-1

CURVES, POLYGONS, AND CIRCLES Curves Triangles and Quadrilaterals Circles Slide 9-2-2

CURVES Slide The basic undefined term curve is used for describing figures in the plane.

SIMPLE CURVE; CLOSED CURVE Slide 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.

SIMPLE CURVE; CLOSED CURVE Slide Simple; closed Simple; not closed Not simple; closed Not simple; not closed

CONVEX Slide 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 Slide 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.

POLYGONS Slide Regular Polygons ConvexNot convex

CLASSIFICATION OF POLYGONS ACCORDING TO NUMBER OF SIDES Slide Number of SidesName 3Triangle 4Quadrilateral 5Pentagon 6Hexagon 7Heptagon 8Octagon 9Nonagon 10Decagon

TYPES OF TRIANGLES - ANGLES Slide All Angles Acute One Right Angle One Obtuse Angle Acute TriangleRight TriangleObtuse Triangle

TYPES OF TRIANGLES - SIDES Slide All Sides EqualTwo Sides Equal No Sides Equal Equilateral Triangle Isosceles Triangle Scalene Triangle

TYPES OF QUADRILATERALS Slide A rectangle is a parallelogram with a right angle. A trapezoid is a quadrilateral with one pair of parallel sides. A parallelogram is a quadrilateral with two pairs of parallel sides.

TYPES OF QUADRILATERALS Slide 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 Slide The sum of the measures of the angles of any triangle is 180°.

EXAMPLE: FINDING ANGLE MEASURES IN A TRIANGLE Slide Find the measure of each angle in the triangle below. (x + 20)° x°x° (220 – 3x)° Solution x + x – 3x = 180 –x = 180 x = 60 Evaluating each expression we find that the angles are 60°, 80° and 40°.

EXTERIOR ANGLE MEASURE Slide The measure of an exterior angle of a triangle is equal to the sum of the measures of the two opposite interior angles 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 Slide Find the measure of the exterior indicated below. (x + 20)° x°x° (3x – 40)° Solution x + x + 20 = 3x – 40 2x + 20 = 3x – 40 x = 60 Evaluating the expression we find that the exterior angle is 3(60) – 40 =140°.

CIRCLE Slide 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 Slide 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 Slide P R O T Q 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 Slide 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.