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9.2 – Curves, Polygons, and Circles. Curves. The basic undefined term curve is used for describing non-linear figures a the plane. A simple curve can be drawn without lifting the pencil from the paper, and without passing through any point twice.

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Curves

The basic undefined term curve is used for describing non-linear figures a the plane.

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; closed

Simple; not closed

Not simple; closed

Not simple; not closed

Polygons

A polygon is a simple, closed curve made up of only straight line segments.

The line segments are called sides.

The points at which the sides meet are called vertices.

Polygons with all sides equal and all angles equal are regular polygons.

Regular Polygons

Polygons

A figure is said to be convex if, for any two points A and Binside the figure, the line segment AB is always completely inside the figure.

E

F

B

A

M

C

N

D

Convex

Not convex

Classification of Polygons According to Number of Sides

Tridecagon

Icosagon

Hendecagon

Triacontagon

Dodecagon

Tetracontagon

Types of Triangles - Angles

All Acute Angles

One Right Angle

One Obtuse Angle

Acute Triangle

Right Triangle

Obtuse Triangle

Types of Triangles - Sides

All Sides Equal

Two Sides Equal

No Sides Equal

Equilateral Triangle

Isosceles Triangle

Scalene Triangle

Quadrilaterals: any simple and closed four-sided figure

A trapezoid is a quadrilateral with one pair of parallel sides.

A parallelogram is a quadrilateral with two pairs of parallel sides.

A rectangle is a parallelogram with a right angle.

A square is a rectangle with all sides having equal length.

A rhombus is a parallelogram with all sides having equal length.

Triangles

Angle Sum of a Triangle

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

Find the measure of each angle in the triangle below.

(220 – 3x)°

(x+20)°

x + x + 20 + 220 – 3x = 180

x = 60°

–x + 240 = 180

60+ 20 = 80°

– x = – 60

220 – 3(60) = 40°

x = 60

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.

2

Exterior angle

4

1

3

The measure of angle 4 is equal to the sum of the measures of angles 2 and 3.

m4 = m2 + m3

Find the measure of the exterior indicated below.

(x+20)°

(3x – 50)°

(x – 50)°

3x – 50 = x + x + 20

3x – 50 = 2x + 20

3x = 2x + 70

x = 70

3(70) – 50

160°

Circles

A circle is a set of points in a plane, each of which is the same distance from a fixed point (called the center).

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.

A portion of the circumference of a circle between any two points on the circle is called an arc.

O is the center

P

PQ is an arc.

M

LM is a chord.

Q

O

PQ is a secant line.

PR is a diameter.

(PQ is a chord).

L

T

R

RT is a tangent line.

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.

Congruent Triangles

Congruent triangles: Triangles that are both the same size and same shape.

E

B

A

D

C

F

The corresponding sides are congruent.

The corresponding angles have equal measures.

Notation:

Using Congruence Properties

Side-Angle-Side (SAS) If two sides and the included angle of one triangle are equal, respectively, to two sides and the included angle of a second triangle, then the triangles are congruent.

B

E

A

D

C

F

Using Congruence Properties

Angle-Side-Angle (ASA) If two angles and the included side of one triangle are equal, respectively, to two angles and the included side of a second triangle, then the triangles are congruent.

E

B

A

D

C

F

Congruence Properties - SSS

Side-Side-Side (SSS) If three sides of one triangle are equal, respectively, to three sides of a second triangle, then the triangles are congruent.

B

E

A

D

C

F

Proving Congruence

CE = ED

CE = ED

Given:

C

B

AE = EB

AE = EB

E

A

Prove:

ACE  BDE

D

Given

1.

1.

2.

2.

Given

3.

AEC = BED

3.

Vertical angles are equal

ACE  BDE

4.

4.

SAS property

Proving Congruence

B

Given:

C

ABD = CDB

A

D

Prove:

ABD  CDB

Given

1.

1.

2.

ABD = CDB

2.

Given

3.

BD = BD

3.

Reflexive property

4.

4.

ABD  CBD

ASA property

Proving Congruence

B

Given:

AB = CB

AB = CB

BD = BD

Prove:

ABD  CBD

A

C

D

Given

1.

1.

2.

2.

Given

3.

3.

Reflexive property

4.

4.

ABD  CBD

SSS property

Isosceles Triangles

If ∆ABC is an isosceles triangle with AB = CB, and if D is the midpoint of the base AC, then the following properties hold.

B

1. The base angles A and C are equal.

2. Angles ABD and CBD are equal.

3. Angles ADB and CDB are both right angles.

A

D

C

Similar Triangles:Triangles that are exactly the same shape, but not necessarily the same size.

For triangles to be similar, the following conditions must hold:

1. Corresponding angles must have the same measure.

2. The ratios of the corresponding sides must be constant. That is, the corresponding sides are proportional.

Angle-Angle Similarity Property

If the measures of two angles of one triangle are equal to those of two corresponding angles of a second triangle, then the two triangles are similar.

B

E

8

16

F

24

C

D

32

A

Find the length of sides DF.

Set up a proportion with corresponding sides:

DF = 16.

Pythagorean Theorem

c (hypotenuse)

leg a

leg b

If the two legs of a right triangle have lengths a and b, and the hypotenuse has length c, then

(The sum of the squares of the lengths of the legs is equal to the square of the hypotenuse.)

Find the length a in the right triangle below.

39

a

36

Converse of the Pythagorean Theorem

If the sides of lengths a, b, and c, where c is the length of the longest side, and if

then the triangle is a right triangle.

Is a triangle with sides of length 4, 7, and 8, a right triangle?

Is a triangle with sides of length 8, 15, and 17, a right triangle?

Not a right triangle.

right triangle.