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 9-2-3

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

SIMPLE CURVE; CLOSED CURVE

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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

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Simple; closed

Simple; not closed

Not simple; closed

Not simple; not closed

CONVEX

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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

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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 9-2-8

Regular Polygons

Convex Not convex

CLASSIFICATION OF POLYGONS ACCORDING TO NUMBER OF SIDES

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Number of Sides Name

3 Triangle

4 Quadrilateral

5 Pentagon

6 Hexagon

7 Heptagon

8 Octagon

9 Nonagon

10 Decagon

TYPES OF TRIANGLES - ANGLES

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All Angles Acute

One Right Angle

One Obtuse Angle

Acute Triangle Right Triangle Obtuse Triangle

TYPES OF TRIANGLES - SIDES

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All Sides Equal Two Sides Equal

No Sides Equal

Equilateral Triangle

Isosceles Triangle

Scalene Triangle

TYPES OF QUADRILATERALS

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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 9-2-13

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

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The sum of the measures of the angles of any triangle is 180°.

EXAMPLE: FINDING ANGLE MEASURES IN A TRIANGLE

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Find the measure of each angle in the triangle below.

(x + 20)°

x°(220 – 3x)°

Solution

x + x + 20 + 220 – 3x = 180 –x + 240 = 180

x = 60

Evaluating each expression we find that the angles are 60°, 80° and 40°.

EXTERIOR ANGLE MEASURE

Slide 9-2-16

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

34

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 9-2-17

Find the measure of the exterior indicated below.(x + 20)°

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

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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

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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 9-2-20

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 9-2-21

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.