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.

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

9.2 – Curves, Polygons, and Circles 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

9.2 – Curves, Polygons, and Circles

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.

E F A B C D

Convex

M N

Not convex

9.2 – Curves, Polygons, and Circles

Classification of Polygons According to Number of Sides

Number of Sides

3 4 5 6 7 8 9 10 11 12

Name

Triangle Quadrilateral Pentagon Hexagon Heptagon Octagon Nonagon Decagon Hendecagon Dodecagon

Number of Sides

13 14 15 16 17 18 19 20 30 40

Names

Tridecagon Tetradecagon Pentadecagon Hexadecagon Heptadecagon Octadecagon Nonadecagon Icosagon Triacontagon Tetracontagon

9.2 – Curves, Polygons, and Circles Types of Triangles - Angles

All Acute Angles One Right Angle One Obtuse Angle

Acute Triangle Right Triangle Obtuse Triangle

9.2 – Curves, Polygons, and Circles Types of Triangles - Sides

All Sides Equal Two Sides Equal No Sides Equal

Equilateral Triangle Isosceles Triangle Scalene Triangle

9.2 – Curves, Polygons, and Circles

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.

9.2 – Curves, Polygons, and Circles 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.

x

° (

x

+20)° (220 – 3

x

)°

x

+

x

+ 20 + 220 – 3

x

= 180 –

x

+ 240 = 180

– x

= – 60

x

= 60

x

= 60 ° 60 + 20 = 80° 220 – 3( 60 ) = 40°

9.2 – Curves, Polygons, and Circles 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

9.2 – Curves, Polygons, and Circles

Find the measure of the exterior indicated below.

(3

x –

50)° (

x –

50)° (

x

+20)°

x

° 3

x –

50

= x

+

x

+ 20 3

x

– 50 = 2

x

+ 20 3

x

= 2

x

+ 70

x

= 70 3( 70 ) – 50 160°

9.2 – Curves, Polygons, and Circles 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

.

9.2 – Curves, Polygons, and Circles

O

is the center

OQ

is a radius.

LM

is a chord.

PR

is a diameter.

M P PQ

is an arc.

O L T R RT

is a tangent line.

Q PQ

is a secant line.

(PQ

is a chord).

9.2 – Curves, Polygons, and Circles 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.

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Congruent Triangles

Congruent triangles:

Triangles that are both the same size and same shape.

E B D A F

The corresponding sides are congruent. The corresponding angles have equal measures. Notation: 

ABC

 

DEF

.

C

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Using Congruence Properties

Side-Angle-Side (SAS)

congruent.

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

E B D A F



ABC

 

DEF

.

C

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Using Congruence Properties

Angle-Side-Angle (ASA)

congruent.

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

E B D A F



ABC

 

DEF

.

C

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem 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.

E B D A F



ABC

 

DEF

.

C

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Proving Congruence

Given

:

CE = ED Prove: AE = EB

 ACE   BDE

A C E D B

STATEMENTS REASONS 1.

CE = ED

1.

Given 2.

AE = EB

3.

 AEC =  BED 4.

 ACE   BDE 2.

Given 3.

Vertical angles are equal 4.

SAS property

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

Given

: Proving Congruence 

ADB =



CBD



ABD =



CDB B C Prove:

 ABD   CDB

A D

STATEMENTS 1. 

ADB =



CBD

2.



ABD =



CDB

3.

BD = BD 4.

 ABD   CBD REASONS 1.

Given 2.

Given 3.

Reflexive property 4.

ASA property

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Proving Congruence

B Given

:

AD = CD Prove: AB = CB

 ABD   CBD

A C D

STATEMENTS REASONS 1.

AD = CD

1.

Given 2.

AB = CB

3.

BD = BD

4.

 ABD   CBD 2.

Given 3.

Reflexive property 4.

SSS property

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem 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

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

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.

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem 

ABC

is similar to 

DEF

.

E B

8 16

F

24

C D

32

A

Find the length of sides

DF

.

Set up a proportion with corresponding sides:

EF BC



DF AC

8 16 

DF

32

DF

= 16.

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem 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

a

2 

b

2 

c

2

.

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

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Find the length

a

in the right triangle below.

39

a

36

a

2

a

2 

b

2  36 2 

c

2  39 2

a

2 

a

2  225

a

 15

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem Converse of the Pythagorean Theorem If the sides of lengths

a

,

b

, and

c

, where

c

longest side, and if

a

2 

b

2 

c

2

,

is the length of the then the triangle is a right triangle.

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

4 2  7 2 65    8 2 64 64 Not a right triangle.

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

right triangle.