long lynette groovy geometry games and activiti

In this section, you’ll practice measuring and creating different angles,learn the relationship between some interesting angle pairs, discover therelationship between the angles formed w

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Copyright © 2003 by Lynette Long All rights reserved

Illustrations copyright © 2003 by Tina Cash-Walsh

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The publisher and the author have made every reasonable effort to ensure that the experiments and activities

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ISBN 0-471-21059-5 (pbk : alk paper)

1 Geometry—Study and teaching (Elementary)—Activity programs 2 Games in

22 Parallelogram Presto Change-O 66

Geometry is the study of

points, lines, angles,

and shapes, and their

rela-tionships and properties

It sounds like a lot to

know, but much of it is

already in your head

Geometry is all around

us If people didn’t think

about geometry, they

wouldn’t be able to build

great structures such as the

pyr-amids, or even simple things that

lie flat such as a table

Geometry can be easily learned by experimenting and having fun withthings you can find around the house You can learn most of the principles

of geometry using cereal boxes, soda cans, plates, string, magazines, andother common household objects So get ready to have a great time explor-ing the world of geometry

Plane:a flat surface that extends infinitely in alldirections

Point:a location on a plane

Line:a straight path of points that goes on indefinitely

Line segment:all of the points on a linebetween two specific end points

Ray:all of the points on a line going out fromone end point indefinitely in one direction

Plane geometry:the study of dimensional figures

two-S OME K EY T ERMS TO K NOW

I I

A n angle is formed by the meeting of two rays at the same end point.

The point where the two rays meet is called the angle’s vertex The

rays are called the sides of the angle

Angles are everywhere When you bend your arm, your elbow becomes

the vertex of the angle formed by the two parts of your arm When two

streets cross each other, they form angles Here are someexamples of angles:

Angles are measured indegrees If an angle is lessthan 90 degrees, it is called

an acute angle If it is

exactly 90 degrees, it is

called a right angle And

if it is more than 90degrees, it is called an

obtuse angle.

In this section, you’ll practice measuring and creating different angles,learn the relationship between some interesting angle pairs, discover therelationship between the angles formed when two parallel lines are inter-sected by another line, practice recognizing right angles and perpendicularlines, and more.

Along the way, you’ll measure angles around your house, have an drawing competition, play a game of matching angle pairs, create numbersusing only perpendicular lines, and go on a right-angle scavenger hunt

Angles can be identified by labeling a point

on each ray and the point that is the vertex

For example, the angle

can be written as angle ABC or angle CBA

(note that the vertex point always goes in the

middle) You can also write this angle using

an angle symbol as ∠ABC or∠CBA.

Measure Up

Angles are measured in degrees using a protractor.

If you’ve never used one, don’t worry It’s easy and fun

You just align the bottom marking of the protractor with one

ray of the angle you want to measure The vertex of the angle

should be seen through the hole in the protractor Next, read

the number on the protractor nearest to where the second ray

crosses Your protractor has two sets of numbers The one you

use depends on the starting direction of the angle You need to

figure out which set of numbers has the first ray starting at

zero, then count up from there to find the right number Try

this activity to practice measuring angles with a protractor.

MAT E R I A L S

protractor pencil paper scissors cardboard ruler paper brad

1 Look around any

room in your house

for lines that meet

at corners—for

example, tables,

picture frames,

blocks, books, clock

hands, and so on

2 Use the protractor

to measure some of

the angles created

by the things in the room

5 Cut out two strips from the cardboard that are about 1 inch (2.5 cm) ×8

inches (20 cm) Use a ruler to draw a ray down the middle of each strip

Connect the strips of cardboard at the end points of the two rays using

the paper brad

6 Use the cardboard rays to make different angles and measure the angles

with your protractor

Draw It!

Try this game to practice

drawing angles of different measures.

Game Preparation

1 Cut each sheet of paper into eight small pieces.

2 Write one of the following measurements on each small piece of paper:

3 Fold the pieces of paper so that you can’t see the measurements and place

them in the bowl

Game Rules

1 Player 1 reaches into the bowl and picks a piece of paper Player 1 reads

the number of degrees out loud and tries to draw an angle with this

measure using only a pencil and a ruler

2 Player 2, using a protractor, measures the angle drawn and writes the

measure of the angle inside the angle

3 Player 2 finds the difference between the measure of the angle as noted

on the paper and the actual measure of the angle drawn

4 Player 1 rolls a single die If the difference between the measure of the

angle drawn and the measure on the piece of paper is less than the

num-ber rolled, then Player 1 earns 1 point

EXAMPLE

Player 1 is supposed to draw a 30-degree angle, but when Player 2

measures it using a protractor, the angle is actually 34 degrees The

difference between these two measures is 4 degrees (34 – 30 = 4)

Player 1 rolls a 5 on the die Player 1 earns 1 point, since the

differ-ence of 4 degrees is less than the number rolled, which is 5

5 Player 2 selects a piece of paper from the bowl, reads the number of

degrees out loud, and tries to draw an angle with that measure Player 1

measures the angle drawn with a protractor and writes the measure

Player 2 rolls the die to determine if his or her drawing is accurate enough

to earn a point

6 The first player to earn 3 points wins the game

90 45

s

degree

s

Create new slips of paper and write new angle

measures on them Make the angles measurebetween 0 and 360 degrees Play the game again

using these new measures

Name Game

Play this fast-paced game to practice identifying acute, right,

and obtuse angles (see pages 3–4 for definitions and

Game Rules

1 Deal eight cards to each player.

2 Both players put their cards facedown in a stack in front of them.

3 Players turn over their top cards at the same time and put them down

faceup next to each other

4 Each player calls out at the same time whether his or her angle is acute,

right, or obtuse An obtuse angle beats a right or an acute angle, and aright angle beats an acute angle The winner gets to keep both cards If theangles are both acute or both obtuse, then the largest angle wins If theangles are both right, then the winner of the next round gets to keep thecards If a player calls out the wrong type of angle, then he or she losesthe round no matter what angle is showing on the card

5 When all the cards have been played, the player with the most cards is the

winner

Angle Pairs

Angles also have different names that refer to special

relation-ships that some angles have with other angles These pairs of

angles are called vertical, complementary, and supplementary.

Vertical angles are opposite angles that are formed when two

lines intersect They have the same measurement For

exam-ple, if the original angle is 31 degrees, the vertical angle is

31 degrees Complementary angles are angles

whose measurements add up to 90 degrees.

For example, if the original angle is 12

degrees, the complementary angle

meas-ures 78 degrees (90 – 12 = 78)

Sup-plementary angles are angles whose

measurements add up to 180 degrees.

For example, if the original angle is 25 degrees, the

sup-plementary angle measures 155 degrees (180 – 25 = 155) Play

this game to practice computing the measures of complementary,

supplementary, and vertical angles.

Game Preparation

1 Write the words vertical angles on five index cards Write the words

comple-mentary angles on five index cards Write the words supplecomple-mentary angles on

five index cards

Game Rules

1 Shuffle the cards and deal each player seven cards There should be one

MAT E R I A L S

2 players pencil

15 index cards dice

2 Player 1 rolls the dice and uses the numbers rolled to form a two-digit

number The larger number rolled is the tens place and the smaller ber rolled is the ones place For example, if a 6 and a 4 are rolled, thenumber rolled is 64 This is the number of your original angle

num-3 Players each select one of their index cards and place it faceup on the

table If the cards are the same, Player 2 turns over another card until he

or she gets a different type of angle card

4 Players compute the value of the type of angle named on their cards from

the original angle The player with the larger angle wins both cards

EXAMPLE

The number rolled is 55 Player 1 selected a vertical angle card Thevertical angle of an angle that measures 55 degrees is an angle thatmeasures 55 degrees Player 2 selected a complementary anglecard The complementary angle of an angle that measures 55

degrees is 35 degrees (90 – 55 = 35) Player 1 wins both cards,since 55 degrees is greater than 35 degrees

5 Players take turns rolling the dice and calculating the angles until the

cards have run out The winner is the player with the most cards at theend of the game

Color by Angles

Parallel lines are lines on the same plane

that will never intersect A

transversal is a line that

intersects two parallel lines.

When two parallel lines are

cut by a transversal,

they form eight

angles The angles can

be named according to

their position

Angles between the two parallel lines are

interior angles.

Angles 3, 4, 5, and 6 are interior angles

Angles outside the parallel lines are exterior

angles.

Angles 1, 2, 7, and 8 are exterior angles

Angles on opposite sides of the transversal

that have the same measurement are

Angles 1 and 8 are alternate exterior angles

Angles 2 and 7 are alternate exterior angles

Angles on the same side of the transversal that have the same

measure-ment are corresponding angles.

Angles 1 and 5 are corresponding angles

Angles 2 and 6 are corresponding angles

Angles 3 and 7 are corresponding angles

Angles 4 and 8 are corresponding angles

Try this activity to see the relationship between the angles formed when two parallel lines are intersected by a transversal.

Procedure

1 Using a ruler, draw two parallel lines on a piece of paper.

2 Using a ruler, draw a transversal across the lines

3 Label the eight angles 1, 2, 3, 4, 5, 6, 7, and 8, as in the illustration on

page 15

4 Using a protractor, measure each of the angles Write the measures on a

separate piece of paper

6 Add any two angles with different measures What is the sum of these two

angles?

Any two different angles in the

fig-ure will be supplementary angles,

which means they will always add up

to 180 degrees

90 45

Find the alternate interior

and exterior angles and thecorresponding angles in thepicture you colored

Perpendicular

Numbers

A right angle is an angle of 90 degrees Two lines that meet

in a right angle are called perpendicular lines Try this

activ-ity to form numbers using only perpendicular lines.

Procedure

MAT E R I A L S

highlighter paper toothpicks

2 Use your toothpicks to cover each of the seven segments You have made

the number 8 using only perpendicular lines

3 Now see if you can use the toothpicks to make all the numbers from 0

to 9 using only perpendicular lines (Hint: use the highlighted lines from

the number 8 figure as a base from which to create the rest of the

numbers.)

4 How many right angles can you find in each number?

90 45

Right-Angle Scavenger Hunt

Now that you know what right angles look

like, play this game to discover the right

3 After 10 minutes, players read their lists to each other The player with the

longest list wins the game

90 45

I I I

When you connect straight lines to make a closed two-dimensional

shape, the result is a polygon Polygons include squares, triangles,

and rectangles, which you probably know a lot about But they also

include pentagons (with five sides), hexagons (with six sides), heptagons

(with seven sides), and many more

Polygons with three sides are some of the most interesting figures ingeometry You probably know that these figures are called triangles But tri-angles are not as simple as they first appear There are many types of trian-gles Many of them are named after the types of angles they contain, such asacute, obtuse, and right There are also scalene and isosceles triangles

In this section, you’ll learn about many of the different kinds of triangles,the exterior and interior angles in a triangle, congruent triangles, and thePythagorean theorem

Along the way, you’ll make a triangle collage, play triangle memory, puttogether a triangle puzzle, and use triangles to figure out the heights ofobjects Triangles are fascinating So let’s get started!

There are six basic types of triangles:

1 Acute triangle: All three angles of an

acute triangle are less

than 90 degrees.

2 Obtuse triangle:

One angle of an obtuse

triangle is greater than

90 degrees.

3 Right triangle: One angle

of a right triangle is equal to

90 degrees.

4 Scalene triangle: All three sides of a

sca-lene triangle have different measures.

5 Isosceles triangle: Two angles and

two sides of an isosceles

triangle are equal.

6 Equilateral triangle: All

three angles and all three

sides of an equilateral

triangle are equal.

In this activity, you’ll

make a poster using all

the different triangle types.

8

Triangle Collage

MAT E R I A L S

marker old magazines scissors glue poster board

1 Draw at least six large triangles on different

maga-zine pages Make a picture in the magamaga-zine the

center of each triangle For example, you could

draw a triangle around a person’s face like this:

Draw one of every type of triangle in the

list on page 25

2 Cut out each triangle.

3 Glue the triangles on the poster board in

a way that makes an interesting picture

90 45

s

degree

s

Make another collage out of all right triangles

or all obtuse triangles or all equilateral gles How does that change the way the collage

trian-looks?

Play this game to practice matching

triangles to their type.

2 Write one of the following sets of angle measurements on each index card

of the second color

Game Rules

1 Shuffle the cards of one color together and place them facedown on the

table Do the same for the other color

2 Player 1 turns over one card of each color If the measurements on one

card match the definition of the type of triangle described on the othercard, the player keeps both cards If they are not a match, Player 1 placesthem facedown on the table

3 Player 2 now turns over two cards (one of each color) and tries to find a

match If both cards match, Player 2 gets to keep both cards

4 When there are no cards left or no more matches on the table, the player

with the most cards is the winner

Use the angles of a triangle as the pieces of a

puzzle and learn something interesting

about the sum of a triangle’s angles.

Procedure

1 Use a ruler to draw a triangle on a piece of paper.

2 Cut out the triangle.

3 Rip off the three corners of the triangle.

4 Put the three corners in a row so that the angles meet at one point and at

least one side of each angle touches the side of another angle

5 The angles of the triangle should form a straight line The sum of the

angles of your triangle is 180 degrees

10

Triangle Angles

MAT E R I A L S

pencil paper ruler scissors

6 Draw another triangle and repeat steps 2 through 5.

90 45

60

80

square

es Can you draw a triangle whosedeg

angles do not add up to 180degrees?

Six exterior angles can be formed outside

every triangle An exterior

angle is formed by one side of

a triangle and the extension

of another side (See

angles 1 through 6 in

the figure below.)

Try this activity to

learn more about the

exterior angles of a triangle.

1 Use a ruler to draw an acute triangle Extend one of the sides of the

tri-angle to form an exterior tri-angle

2 Label the exterior angle a Label the interior angles 1, 2, and 3 In the

tri-angle shown here, tri-angle a is the exterior tri-angle, tri-angle 3 is the adjacent

interior angle, and angles 1 and 2 are the nonadjacent interior angles

3 Use a protractor to measure the exterior angle Enter this measurement

in a chart like the one

on page 34

4 Use a protractor to

measure each of the

nonadjacent interior

angles Enter these

meas-urements in your chart

6 Use a ruler to draw a right triangle.

7 Extend one of the sides of the right triangle.

8 Label the exterior angle a Label the interior angles 1, 2, and 3 In the

above triangle, angle a is the exterior angle, angle 3 is the adjacent

inte-rior angle, and angles 1 and 2 are nonadjacent inteinte-rior angles

9 Use a protractor to measure the exterior angle Enter this measurement

in your chart

10 Measure each of the nonadjacent interior angles Enter these

measure-ments in your chart

11 Add the measurements of angles 1 and 2 Enter the result in your chart.

12 Draw an obtuse triangle.

13 Extend one of the sides of the obtuse triangle.

14 Label the exterior angle a Label the interior angles 1, 2, and 3 In the

above triangle, angle a is the exterior angle, angle 3 is the adjacent

inte-rior angle, and angles 1 and 2 are the nonadjacent inteinte-rior angles

15 Measure the exterior angle Enter this measurement in your chart.

16 Measure each of the nonadjacent interior angles Enter these

measure-ments in your chart

17 Add the measurements of angles 1 and 2 Enter the result in your chart.

Exterior angle a should always equal the sum of interior angles 1

and 2—that is, any exterior angle of a triangle is equal to the

sum of the two nonadjacent interior angles (also known as remote

What is the sum of the six