CIRCLES

BASIC TERMS AND FORMULAS

Natalee Lloyd

Basic Terms and Formulas

Terms Center Radius Chord Diameter Circumference

Formulas Circumference

formula Area formula

Center: The point which all points of the circle are

equidistant to.

Radius: The distance from the center to a point on the

circle

Chord: A segment connecting two points on the

circle.

Diameter: A chord that passes through the center of

the circle.

Circumference: The distance around a circle.

Circumference Formula: C = 2r or C = d

Area Formula: A = r2

Circumference Example

C = 2r

C = 2(5cm)

C = 10 cm5 cm

Area Example

A = r2

Since d = 14 cm then r = 7cm

A = (7)2

A = 49 cm

14 cm

Angles in Geometry

Fernando Gonzalez - North Shore High School

Intersecting Lines Two lines that

share

one common point.

Intersecting lines can

form different types of

angles.

Complementary Angles

Two angles that equal 90º

Supplementary Angles

Two angles that equal 180º

Corresponding Angles Angles that are

vertically identical

they share a common vertex and have a line running through them

Geometry

Basic Shapes

and examples in everyday life

Richard Briggs

NSHS

GEOMETRY

Exterior Angle Sum Theorem

What is the Exterior Angle Sum Theorem?

The exterior angle is equal to the sum of the interior angles on the opposite of the triangle.

11070

40

70

110 = 70 +40

Exterior Angle Sum Theorem

There are 3 exterior angles in a triangle. The exterior angle sum theorem applies to all exterior angles.

11664 64

52

116

128

128 = 64 + 64 and 116 = 52 + 64

Linking to other angle concepts

As you can see in the diagram, the sum of the angles in a triangle is still 180 and the sum of the exterior angles is 360.

20

80 80100100

160

80 + 80 + 20 = 180 and 100 + 100 + 160 = 360

Geometry

Basic Shapes

and examples in everyday life

Barbara Stephens

NSHS

GEOMETRY

Interior Angle Sum Theorem

What is the Interior Angle Sum Theorem?

The interior angle is equal to the sum of the interior angles of the triangle.

11070

40

70

110 = 70 +40

Interior Angle Sum Theorem

There are 3 interior angles in a triangle. The interior angle sum theorem applies to all interior angles.

11664 64

52

116

128

128 = 64 + 64 and 116 = 52 + 64

Linking to other angle concepts

As you can see in the diagram, the sum of the angles in a triangle is still 180.

20

80 80100100

160

80 + 80 + 20 = 180

GeometryParallel Lines with a Transversal

Interior and exterior Angles

Vertical Angles

By

Sonya Ortiz

NSHS

Transversal Definition: A transversal is a

line that intersects a set of parallel lines.

Line A is the transversal

A

Interior and Exterior Angles

Interior angels are angles 3,4,5&6.

Interior angles are in the inside of the parallel lines

Exterior angles are angles 1,2,7&8

Exterior angles are on the outside of the parallel lines

1 23 4

5 67 8

Vertical Angles Vertical angles are

angles that are opposite of each other along the transversal line.

Angles 1&4 Angles 2&3 Angles 5&8 Angles 6&7 These are vertical

angles

1 23 45 67 8

Summary Transversal line intersect parallel lines.

Different types of angles are formed from the transversal line such as: interior and exterior angles and vertical angles.

Geometry

Parallelograms

M. Bunquin

NSHS

Parallelograms A parallelogram is a a special

quadrilateral whose opposite sides are congruent and parallel.

D

A B

C

Quadrilateral ABCD is a parallelogram if and only if

1. AB and DC are both congruent and parallel

2. AD and BC are both congruent and parallel

Kinds of Parallelograms Rectangle

Square

Rhombus

Rectangles

Properties of Rectangles 1. All angles measure 90 degrees. 2. Opposite sides are parallel and congruent. 3. Diagonals are congruent and they bisect each

other. 4. A pair of consecutive angles are

supplementary. 5. Opposite angles are congruent.

Squares

Properties of Square 1. All sides are congruent. 2. All angles are right angles. 3. Opposite sides are parallel. 4. Diagonals bisect each other and they are

congruent. 5. The intersection of the diagonals form 4

right angles. 6. Diagonals form similar right triangles.

Rhombus

Properties of Rhombus 1. All sides are congruent. 2. Opposite sides parallel and opposite angles

are congruent. 3. Diagonals bisect each other. 4. The intersection of the diagonals form 4 right

angles. 5. A pair of consecutive angles are

supplementary.

Geometry

Pythagorean Theorem

Cleveland Broome

NSHS

Pythagorean Theorem The Pythagorean theorem This theorem reflects the sum of the

squares of the sides of a right triangle

that will equal the square of the hypotenuse.

C2 =A2 +B2

A right triangle has sides a, b and c.

b

a

c

If a =4 and b=5 then what is c?

Calculations:

A2 + B2 = C2

16 + 25 = 41

To further solve for the length of C

Take the square root of C

41 = 6.4

This finds the length of the Hypotenuseof the right triangle.

The theorem will help calculate distance when travelingbetween two destinations.

GEOMETRY

Angle Sum Theorem

By: Marlon Trent

NSHS

Triangles Find the sum of the

angles of a three sided figure.

Quadrilaterals Find the sum of the

angles of a four sided figure.

Pentagons Find the sum of the

angles of a five sided figure.

Hexagon Find the sum of the

angles of a six sided figure.

Heptagon Find the sum of the

angles of a seven sided figure.

Octagon Find the sum of the

angles of an eight sided figure.

Complete The ChartName of figure Number of

sidesSum of angles

Triangle

Quadrilateral

Pentagon

Hexagon

Heptagon

Octagon

Decagon

n-agon

What is the angle sum formula?

Angle Sum=(n-2)180 Or Angle Sum=180n-360

A presentation by

A SQUARE IS RECTANGLE

THE SQUARE IS A RECTANGLE

OR

THE RECTANGLE IS A SQUARE

SQUARE Characteristics:Four equal sidesFour Right Angles

RECTANGLE Characteristics

Opposite sides are equal Four Right Angles

Square and Rectangle share

Four right angles Opposite sides are equal

SQUARE AND RECTANGLE DO NOT SHARE:

All sides are equal

SO A SQUARE IS RECTANGLE A RECTANGLE IS NOT A SQUARE

Charles Upchurch

Types of Triangles

Triangles Are Classified Into 2 Main Categories.

Triangles Classified by Sides

Triangles Classified by Their Sides

Scalene Triangles

These triangles have all 3 sides of different lengths.

Isosceles Triangles These triangles have at least 2 sides

of the same length. The third side is not necessarily the same length as the other 2 sides.

Equilateral Triangles These triangles have all 3 sides of the

same length.

Triangles Classified by their

Angles

Acute Triangles

These Triangles Have All Three Angles That Each Measure Less

Than 90 Degrees.

Right Triangles

These triangles have exactly one angle that measures 90 degrees. The other 2 angles will each be

acute.

ObtuseTriangles

These triangles have exactly one obtuse angle, meaning an angle greater than 90 degrees, but less than 180 degrees. The other 2 angles will each be acute.

Paulette Granger

Quadrilaterals

A polygon that has four sides

Quadrilateral Objectives Upon completion of this lesson, students

will: have been introduced to quadrilaterals and

their properties. have learned the terminology used with

quadrilaterals. have practiced creating particular

quadrilaterals based on specific characteristics of the quadrilaterals.

Parallelogram• A quadrilateral that

contains two pairs of parallel sides

                                          

 

Rectangle• A parallelogram with

four right angles

Square• A parallelogram with

four congruent sides and four right angles

Group Activity

Each group design a different quadrilateral and prove that its creation fits the desired characteristics of the specified quadrilateral. The groups could then show the class what they created and how they showed that the desired characteristics were present.

Geometry

Classifying Angles

Dorothy J. Buchanan--NSHS

Right angle90°

Straight Angle180°

Examples

Acute angle35°

Obtuse angle135°

If you look around you, you’ll see angles are everywhere. Angles are measured in degrees. A degree is a fraction of a circle—there are 360 degrees in a circle, represented like this: 360°.

You can think of a right angle as one-fourth of a circle, which is 360° divided by 4, or 90°.

An obtuse angle measures greater than 90° but less than 180°.

Complementary & Supplementary

Angles

Olga Cazares

North Shore High School

Complementary AnglesComplementary

angles are two adjacent angles whose sum is 90°

30 °

60 °

60 ° + 30 ° = 90°

Supplementary AnglesSupplementary

angles are two adjacent angles whose sum is 180°120°

60°

120° + 60° = 180°

ApplicationFirst look at the picture.

The angles are complementary angles.

Set up the equation:

12 + x = 180

Solve for x:

x = 168°

12°

x

Right Anglesby

Silvester Morris

RIGHT ANGLES RIGHT ANGLES

ARE 90 DEGREE ANGLES.

STREET CORNERS HAVE RIGHT ANGLES

SILVESTER MORRIS NSHS

Parallel and Perpendicular Lines

byMelissa Arneaud

Recall: Equation of a straight line: Y=mX+C Slope of Line = m Y-Intercept = C

Parallel Lines Symbol: “||”

Two lines are parallel if they never meet or touch.

Look at the lines below, do they meet?

Line AB is parallel to Line PQ or AB || PQ

Slopes of Parallel Lines If two lines are parallel then they have

the same slope.

Example:

Line 1: y = 2x + 1

Line 2: y = 2x + 6

THINK: What is the slope of line 1?

What is the slope of line 2?

Are these two lines parallel?

Perpendicular Lines Two lines are perpendicular if they

intersect each other at 90°.

Look at the two lines below: A

BC

D

Is AB perpendicular to CD? If the answer is yes, why?

Slopes of Perpendicular Lines

The slopes of perpendicular lines are negative reciprocals of each other.

Example:Line 3: y = 2x + 5Line 4: y = -1/2 x + 8THINK: What is the slope of line 3?What is the slope of line 4?Are these two lines perpendicular. If so, why?Show your working.

What do you need to knowParallel Lines

1. Do not intersect.

2. If two lines are parallel then their slopes are the same.

Perpendicular Lines

1. Intersect at 90°(right angles).

2. If two lines are perpendicular then their slopes are negative reciprocals of each other.

Questions1. Write an equation of a straight line that is

parallel to the line y = -1/3 x + 7

State the reason why your line is parallel to that of the line given above.

2. Write an equation of a straight line that is perpendicular to the line y = 4/5 x + 3.

State the reason why the line you chose is perpendicular to the line given above.

Basic Shapesby

Wanda Lusk

Basic Shapes

Two Dimensional•Length•Width

Three Dimensional•Length•Width•Depth (height)

Basic ShapesTwo Dimensions

•Circle•Triangle•Parallelogram

• Square• Rectangle

Basic ShapesTwo Dimensions

•Circle

Basic ShapesTwo Dimensions

•Triangle

Basic ShapesTwo Dimensions

•Square

Basic ShapesTwo Dimensions

•Square•Rectangle

Basic ShapesThree Dimensions

•Sphere•Cone•Cube•Pyramid•Rectangular Prism

Basic ShapesThree Dimensions

•Sphere•Cone•Cube•Pyramid•Rectangular Prism