9.2 – Curves, Polygons, and Circles

9.2 – Curves, Polygons, and CirclesCurves

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

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

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

The line segments are called sides.

The points at which the sides meet are called vertices.

Polygons

Regular PolygonsPolygons

9.2 – Curves, Polygons, and CirclesA 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

E F

Convex Not convex

B

D

C

M N

9.2 – Curves, Polygons, and CirclesClassification of Polygons According to Number

of Sides

Number of Sides

Name Number of Sides

Names

3 Triangle 13

4 Quadrilateral 14

5 Pentagon 15

6 Hexagon 16

7 Heptagon 17

8 Octagon 18

9 Nonagon 19

10 Decagon 20

11 30

12 40

Hendecagon

Dodecagon

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 CirclesQuadrilaterals: any simple and closed four-sided figure

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.

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

The sum of the measures of the angles of any triangle is 180°.Angle Sum of a Triangle

Triangles

Find the measure of each angle in the triangle below.

(x+20)°

x°

(220 – 3x)°

x + x + 20 + 220 – 3x = 180

–x + 240 = 180

– x = – 60

x = 60

x = 60°

60 + 20 = 80°

220 – 3(60) = 40°

9.2 – Curves, Polygons, and CirclesExterior 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

34

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

m4 = m2 + m3

Exterior angle

9.2 – Curves, Polygons, and CirclesFind the measure of the exterior indicated below.

3x – 50 = x + x + 20

(x+20)°

x°(3x – 50)° (x – 50)°

3x – 50 = 2x + 20

x = 70

3x = 2x + 70

3(70) – 50

160°

9.2 – Curves, Polygons, and CirclesCircles

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 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.

A diameter divides a circle into two equal semicircles.

9.2 – Curves, Polygons, and Circles

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.M

L

LM 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

A

B

C

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

D

E

FThe corresponding sides are congruent.

.ABC DEF

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.

A

B

CD

E

F

.ABC DEF

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

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.

A

B

C

D

E

F

.ABC DEF

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

A

B

CD

E

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.

.ABC DEF

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

Proving Congruence

Given:

A

B

D

E

C

Given

Prove:

CE = ED

AE = EB

ACE BDE

STATEMENTS REASONS

CE = ED

AE = EB Given

ACE BDE

AEC = BED Vertical angles are equal

SAS property

1. 1.

2. 2.

3. 3.

4. 4.

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

Proving Congruence

Given:

Given

Prove:

ADB = CBD

ABD CDB

STATEMENTS REASONS

Given

Reflexive property

ASA property

1. 1.

2. 2.

3. 3.

4. 4.

ABD = CDB

ADB = CBD

ABD = CDB

ABD CBD

A

B

D

C

BD = BD

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

Proving Congruence

Given:

Given

Prove: ABD CBD

STATEMENTS REASONS

Given

Reflexive property

SSS property

1. 1.

2. 2.

3. 3.

4. 4.ABD CBD

A

B

D C

AD = CD

AB = CB

AD = CD

AB = CB

BD = BD

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.

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 C

B

D

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.

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.

Angle-Angle Similarity Property

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

is similar to .ABC DEF

Find the length of sides DF.

A

B

C

D

E

F16

24

32

8

Set up a proportion with corresponding sides:

EF DF

BC AC

8

16 32

DF DF = 16.

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

Pythagorean Theorem

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

2 2 2.a b c

leg b

leg ac (hypotenuse)

(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.

2 2 2a b c

39

36

a

2 2 236 39a 2 1296 1521a

2 225a 15a

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 is the length of the longest side, and if 2 2 2 ,a b c then the triangle is a right triangle.

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

2 2 24 7 8 16 49 64

65 64

Not a right triangle.

9.3 –Triangles: Congruence, Similarity and the Pythagorean Theorem

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

right triangle.