Geometry

Points, Rays, and Lines

This is a point. A point represents one place in space.

This is a ray. A ray is a straight line of points that begins at one point and goes in one direction.

This is a line. A line is a straight path of points that goes in two directions.

This is a line segment. A line segment is a straight line that begins and ends on two points.

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More Lines

These lines are parallel. They run in the same direction and never cross.

These lines are perpendicular. They cross at right angles.

This is a horizontal line.

This is a vertical line.

This is the point of intersection.Go On

MatchingMatch the symbol on the left with its name on the right.

Horizontal line

Line segment

Point

ray

Vertical line

Perpendicular lines

Parallel lines

Line Go On

Angles

90

90

90

This is an angle. It is formed by two rays extending from the same point.

This is the vertex. It is where the two rays in an angle begin.

This is an acute angle. It measures less than .

This is a right angle. It measures exactly .

This is an obtuse angle. It measures more than but less than . 180

This is a straight angle. It measures exactly . 180

This is a reflex angle. It measures greater than . 180

A degree is a unit which angles are measured.

Go On

MatchingMatch the angle on the left with its name on the right.

Reflex angle

Right angle

Vertex

Straight angle

Acute angle

Obtuse angle Go On

Angles are Everywhere

Acute angle

Right angle

Obtuse angle

Go On

More Lines and Angles

acuteobtuse

acuteobtuse

reflexright

Perpendicular lines

Parallel lines

Go On

Lines and Angles in Real LifeMatch the words in the center with their examples on the

left or right sides.

Perpendicular lines

Parallel lines

Reflex angle

Obtuse angle

Right angle

Acute angle

11

Go On

Angle Relationships

This symbol represents an

angle.

An angle’s name has the vertex as its center. This angle’s name is

ABC.

A

BC

D

E

F

G

W

X

Y

Z

Supplementary Angles Supplementary Angles are two angles that add up to 180°.

For example, in the figure below,

DEF + FEG = 180°.

Complementary Angles Complementary Angles are two

angles that add up to 90°.

For example, in the figure below,

WXY + YXZ = 90°.

Go On

H I J

K L

O P

Q

R ST

U

V

Transversal Line

A Transversal line is a line that crosses parallel lines. In the figure

below, the line with points TSPQ is a transversal line.

Congruent

Congruent means equal. When a transversal line cuts through two parallel lines, the alternate interior and alternate

exterior angles that are formed are congruent.

The following angles are congruent:

OPQ RSP QPV PSU

TSU SPV RST OPS

SPV TSU

Adjacent Angles

Adjacent angles are two angles that share a side.

For example, in the figure below, HIJ and JIL and

HIK and KIL are adjacent angles.

Vertical Angles

The angles that are across from each other (opposite) when two

lines intersect each other are called Vertical Angles. Vertical angles (opposite angles) are equal. The

following angles are vertical angles:

HIK JIL HIJ KIL Go On

Measuring Complementary and Supplementary Angles

CBD? is what ,30 ABC If

ary.complement are CBD and ABC

When you are using both complementary and supplementary angles, it is important to remember that if you know the measurement of one of the angles,

you can find the measurement of the other angle. Remember that Complementary Angles are two angles which add up to 90º. Supplementary

angles are two angles which add up to 180º.

A

B

C

D

30°

Y

W X Z

65°

YXZ? is what ,65 WXY If

ary.supplement are YXZ and WXY

60° 115°

Click Here for a

Problem

Click Here for Answer

Go On

Basic Geometric Shapes

Square A Square is made up of 4 right angles (90° angles) and has 4 sides which are equal in length.

Rectangle A rectangle is a shape that is made up of 4 right angles. The sides

opposite each other are parallel and equal in length.

parallel

parallel Go On

More ShapesTriangle A Triangle is a shape that has three sides. The 3

angles that make up a triangle equal 180°.

Circle A Circle is a shape in which all points that make up the circle are exactly the same distance from the center of the circle. The distance across the center of the circle, which cuts the circle in 2 equal parts, is called the diameter. The distance from the center of the circle to a point on the outside of the circle is the radius.

Right Triangle A Right Triangle is a shape that has 3 sides. One angle (a right angle) is 90°; the right

angle plus the other 2 angles equals 180°. The side opposite the right angle is called the hypotenuse. The other 2 sides are called legs.

radius

diameter

Legs

hypotenuse

Go On

Shapes Are All Around Us

Square

Triangle

RectangleRight Triangle

Circles

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The Pythagorean Theorem

a

b

c

The Pythagorean Theorem states that the square of the hypotenuse is equal to the sum of the squares of the other two sides. The formula

for finding the length of the sides of any right triangle is as follows:

222 bac

Geometry is used in many different occupations by carpenters, architects,

graphic artists, and engineers, just to name a few. It is used in designs, construction, and drawings. Many of these workers need to

know the Pythagorean Theorem.

Go On

Using the Pythagorean Theorem

3

4

c

5

25

25

169

43

2

2

222

222

c

c

c

c

c

bac

Go On

Test Yourself1. What shape is a floor tile that measures 6 inches by 6 inches?

Rectangle Triangle Circle Square

2. If your bedroom measures 15 feet wide and 10 feet long, what is the shape of your bedroom?

Square Triangle CircleRectangle

3. A pizza has a diameter of 12 inches. What shape is your pizza?

Square Rectangle TriangleCircle

Squares Rectangles Circles

4. What shape are these objects?

Triangles

Go On5. A Right Triangle has legs 5 and 12 inches long. How long is the hypotenuse?

17 inches 34 inches 25 inches13 inches

PolygonsPolygons are closed figures (all the segments meet) with three or more sides.

Octagon Hexagon

Pentagon

Triangle

Triangle

Rectangle

Rhombus

Trapezoid

Diamond Square

Shapes with four sides and only four sides are called quadrilaterals. A Rectangle, Rhombus, Parallelogram, Trapezoid,

Diamond, and Square are all quadrilaterals.

A Rhombus is also called a Parallelogram.

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Click on the correct answer.A Quadrilateral shape has ______ sides. One Five Three Four

The sides of a Square are Different Long DiagonalEqual

A shape with three sides is called a Square Rhombus CircleTriangle

How many sides are in a pentagon? One Two ThreeFive

Which of these is not a parallelogram? Rectangle Square Rhombus Hexagon

Which of these is not a quadrilateral? Rectangle Square RhombusTriangle

A parallelogram has opposite sides that are Equal Short Long Parallel

The distance across the center of a circle is called the Radius Length WidthDiameter

Go On

Perimeter and Area of Polygons

30510510

Perimeter is the distance around the outside of a shape. For example, to find the Perimeter of a Rectangle below, add the lengths of all four sides together.

10

10

55

The Perimeter of the Rectangle is 30.

Area is the measure of the space within the perimeter of a shape. To find the Area of the Rectangle below, multiply one length by one width.

50Area

510Area

Area

wl

10

10

55

The Area of the Rectangle is 50.

Go On

Area and Perimeter of Triangles

heightbase2

1

221066

Just like other polygons, to compute the perimeter of a Triangle, you add the sides together.

10

6 6

To compute the area of a Triangle, you multiply

The Perimeter of this triangle is 22.

6 67

10

357102

1

The Area of this triangle is 35.Go On

Circles

4.311014.3

1010

The distance around the outside of a circle is called the Circumference. To find the circumference of a circle, you use a symbol called pi ( ). The formula for the circumference of a circle is •diameter. The value of 14.3

The circumference of this circle is 31.4.

To find the area of a circle, use this formula: (r = radius). Remember, the radius of a circle is half of its diameter.

2r

area5.78

area2514.3

area25

area 52

10

5

diameter

radius

Go On

Practice – Find the Perimeter or Circumference of each shape.

1622222222 5.175.35.35.35.35.3

12354

2 2

2

2

2 2

2

2

3.5 3.5

3.5 3.5

3.5

4

3

5

10

4.3114.310

1010

Click for the Problem

Click for the Solution

Go On

Find the Area of each Shape.

933

66116

2r

3

3 11

6

6

6

8

hb2

1

area113.04

area633.14

area614.3 2

area24

area2

68

area682

1

Click for the Problem

Click for the Solution

Go On

Three-Dimensional Solids

Cube

Cylinder

Rectangular Solid

Cone

Sphere

Go On

Three-Dimensional Solids

Cone

Cube Sphere

Cylinder Rectangular Solid Go On

hwl

V

heightwidthlengthVCube

For a cube or rectangular solid, use the formula

hr

2

2

V

heightRadiusV

For a cylinder, use this formula:

Volume is the measure of space inside a three-dimensional

(not flat) figure.

Volume

Go On

Rectangular Solid

Cylinder

Volume Formulas

heightradius3

1V 2

To find the volume of a cone, use this formula:

3radius3

4V

To find the volume of a sphere, use this formula:

Go OnSphere

Cone

PracticeClick the correct answer.

3in 350

A soccer cone is 8 inches high, with a radius of 5 inches. What is its volume?

8 5

hr3

1V 2

333 in 110 in 700 in 206

A box is 10 inches long by 5 inches wide by 7 inches tall. What is its volume?

105

7

333 in 1047 in 251 in 2863in 209 Go On

PracticeA pair of fuzzy dice have sides which are 4 inches each. What is the volume of one die?

3in 64 33 in 12 in 16

44

4

3in 24

What is the volume of a can of soup that has a radius of 10 cm and a height of 20 cm?

hV 2 r

20

10

33 cm 2000 cm 628 3cm 6280 3cm 314Go On

ReviewHorizontal Line

Vertical Line

Parallel Lines

Perpendicular Lines

Line segment

Ray

Acute Angle

Right Angle

Obtuse Angle

Straight Angle

Reflex Angle

Go On

Review

Supplementary Angles add up to 180°

Complementary Angles add up to 90°

Adjacent Angles are two angles that share a side.

A Transversal Line is a line that crosses parallel lines.

Vertical Angles (Opposite Angles) are Congruent.

Congruent means equal.

Go On

Review

radiusdiameter

Legs

hypotenuse

Octagon Hexagon

Right Triangle

Rectangle

Rhombus

Trapezoid

Diamond Square

Circle

A Quadrilateral Polygon has four sides.

A Polygon is a figure with three or more sides.Pentagon

Triangle

Go On(A Circle is not a polygon.)

The Pythagorean Theorem is222 bac

Review

To find the Area of a Polygon, multiply its length by its width.

To find the Area of a Circle, use this formula: where r is the radius.

To find the Area of a Triangle, multiply its height by half of its base:

wl Area2Area r

hb2

1Area

To find the Perimeter of a Polygon, add all of its sides together.

To find the Circumference of a Circle, multiply the diameter by (3.14).

Diameter

Radius

h

b

Go On

hwl VUse this formula to find the Volume of a Cube or Rectangular Solid:

Use this formula to find the Volume of a Cylinder:

Use this formula to find the Volume of a Cone:

Use this formula to find the Volume of a Sphere:

hr 2V heightradius

3

1V 2

3radius3

4V

Keep Practicing!