Download An Angle on Geometry

Activity Book
An Angle on Geometry
Teacher Edition
GRADES 5-6
For Ages 10+
An Angle on
Geometry
An introduction to geometry,
angles, triangles, circles
and other 2D shapes.
Written by Jane Bourke. Illustrated by Melinda Parker.
ISBN 978-1-63212-037-3
ISBN
978-1-63212-037-3
Phone: 800-507-0966
Fax: 800-507-0967
www.newpathlearning.com
NewPath Learning Products are developed by teachers using research-based principles and are classroom tested. The company’s product line
consists of an array of proprietary curriculum review games, workbooks, posters and other print materials. All products are supplemented with
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applicable state and national educational standards.
Copyright © 2007 Ready-Ed Publications. All Rights Reserved.
Printed in the United States of America
Copyright Notice - Teacher Reproducible Edition
Permission is granted for the purchaser to photocopy sufficient copies for non-commercial educational purposes. However, this permission is not
transferable and applies only to the purchasing individual or institution.
CONTENTS
Teachers’ Notes ..............................................................4
Looking at Different Angles............................................5
Measuring Angles 1 ........................................................6
Reflex Angles..................................................................7
Angles on the Line ..........................................................8
Which Angle is Larger? ..................................................9
Naming Angles ............................................................10
Measuring Angles 2 ......................................................11
Angles in a Triangle......................................................12
Angle Facts ..................................................................13
Scalene Triangles ..........................................................14
Isosceles Triangles ........................................................15
Equilateral Triangles ....................................................16
Get the Right Angle ....................................................17
Angling Around............................................................18
Intersecting Lines..........................................................19
Parallel Lines ................................................................20
Degrees in a Circle ........................................................21
Constructing Angles 1 ..................................................22
Constructing Angles 2 ..................................................23
Angles and Directions 1................................................24
Angles and Directions 2................................................25
Billiards Angles ............................................................26
Angle Check Point........................................................27
Baseball Hits ................................................................28
An Angle on Time ........................................................29
Angles in the Real World ............................................30
Puzzles With Angles ....................................................31
Parts of a Circle 1..........................................................32
Parts of a Circle 2..........................................................33
Triangles in Circles 1 ....................................................34
Triangles in Circles 2 ....................................................35
Angles in Circles 1........................................................36
Angles in Circles 2........................................................37
Shapes in Circles 1 ........................................................38
Shapes in Circles 2 ........................................................39
Answers ........................................................................40
Page 3
TEACHERS’ NOTES
This book is designed to complement the geometry component of the space math strand of the curriculum. It
provides a basic introduction to new concepts as well as activities that will consolidate the skills and ideas
associated with introductory geometry.
The book is designed to be used sequentially as certain skills need to be mastered in order to complete some
of the later activities. Many of the activity pages explain the various mathematical concepts and
provide examples; however, it is assumed that these ideas will be discussed in class prior to students
completing the worksheets.
The activities in this book cover the major learning areas such as identifying different types of angles, using a
protractor to measure angles, using known rules to calculate the size of angles and constructing angles using
either a compass or a protractor.
Angles in a wide range of 2D objects are explored, specifically, the angles of scalene, isosceles and equilateral
triangles, parallel and intersecting lines and angles in a circle. In addition, there are several pages that apply
many of these concepts to angles in everyday situations.
The book also explores the mathematics of circles examining features such as chords, arcs, angles and
various shapes in circles.
Additional materials:
Before starting this unit of work, ensure that each student has access to a compass, a protractor and a ruler. It
is probably best to use pencils rather than pens for construction activities.
Important notes about diagrams:
Occasionally some angles may not appear to be what the answers specify. This is due to slight variations in
the printing process and, unfortunately, these differences are beyond our control.
Rays in diagrams would normally have arrow-heads but they have been omitted in this book to allow
more room.
Also, many 90˚ angles have not been marked with squares to allow diagrams to be more clear. Angles that
look 90˚ generally are 90˚ such as those on pages 7, 14 and 31.
Page 4
LOOKING AT DIFFERENT ANGLES
An angle is the amount of turn between
two lines around a common point.
The lines are known as rays and the point
at which they meet is called a vertex.
A right angle is an angle that measures exactly 90˚
They are often marked with a square at the angle.
An acute angle is an angle less than 90˚. Draw two more examples below.
An obtuse angle measures between 90˚ and 180˚. Draw two more examples below.
❏ Tick the angles below that are right angles. Draw a circle around the acute angles and put a cross inside
the angles that are obtuse.
1.
4.
2.
3.
5.
7.
6.
8.
9.
Page 5
MEASURING ANGLES 1
Angles are measured in degrees. This is usually
expressed with this symbol ˚. A protractor
is used to measure angles.Using a protractor
follow the example below and then complete
the activities.
0
To measure an angle:
1. Place the center of the protractor on the corner or sharpest point (vertex) of the angle.
2. Turn the protractor so that the base line runs along one of the lines that forms the angle.
3. You can then read the size of the angle from the position of the second line. For example this angle is
approximately _______˚
Most protractors number the angles both
clockwise and counter-clockwise.
Make sure that you start at 0 and follow
the correct set of numbers.
❏ Measure the angles below and write down the type of angle for each one, such as acute, obtuse or right.
a.
size: ......................
type: ....................
d.
size: ......................
type: ....................
g.
b.
c.
size: ......................
size: ......................
type: ....................
type: ....................
e.
size: ......................
type: ....................
h.
f.
size: ......................
type: ....................
i.
size: ......................
size: ......................
size: ......................
type: ....................
type: ....................
type: ....................
Page 6
REFLEX ANGLES
A reflex angle is an angle between 180˚ and 360˚. The reflex angle below measures 320˚
❏ 1. Without using a protractor
find the size of the reflex angles below.
a
b
❏ Find the size of the angles below by looking at the size of the reflex angle.
d
e
c
f
180˚ angles
What does a 180˚ degree angle look like?
180˚ degree angles are in fact straight lines.
360˚ angles
This is the full way around the circle. In a circle all angles drawn will always add up to 360˚.
❏ Using a protractor find the size
of the angles in the circle below.
a = ............˚
b = ............˚
c = ............˚
d = ............˚
e = ............˚
Now check if all the angles add up to 360˚.
Page 7
ANGLES ON THE LINE
Supplementary angles are two angles that add up to a total of 180˚.
For example 120˚ + 60˚ = 180˚.
❏ Find the size of the angles marked below.
Complementary angles are two angles that add up to a total of 90˚.
Note: All angles are contained within right angles (90˚).
❏ Fill in the size of the missing angles below.
❏ Using the rules, calculate the size of the supplementary and complementary angles.
Page 8
WHICH ANGLE IS LARGER?
❏ Look at the angles below. Without using a protractor, circle the letter of the largest angle.
1.
2.
3.
4.
5.
6.
7.
8.
9.
❏ Using a protractor find the actual angle size for each of the angles above.
1. A = ..........................
C = ..............................
2. O = ..........................
K = ..............................
3. W = ..........................
N = ..............................
4. G = ..........................
S= ................................
5. R = ..........................
Q = ..............................
6. Z = ..........................
U = ..............................
7. E = ..........................
V = ..............................
8. N = ..........................
P = ................................
9. T = ..........................
K = ................................
What word do the letters of the largest angles in each box spell out? ..........................................
Write a definition for this word ................................................................................................................
Page 9
NAMING ANGLES
Q
This angle has been labelled ∠QRS.
The angle can also be known as angle R or ∠R.
R
❏ Label the angles in each diagram below.
S
......................................
......................................
......................................
......................................
M
Angles in Triangles
Triangles have three angles.
This triangle has been labelled ΔMNO.
The angles can be known as ∠M, ∠N, ∠O.
Note: The triangle is known as MNO rather
than ONM, in keeping with alphabetical order.
O
N
❏ Label the triangles below and then write the name of the largest angle underneath.
C
E
D
e.g. This is ΔCDE
......................................
......................................
......................................
Largest angle is ∠C
......................................
......................................
......................................
Angles in Diagrams
❏ In the diagram below all the points have been labelled.
F
G
M
In a diagram such as this, angles are not referred to by just
one letter such as ∠J.
Instead this angle is known as ∠KJM. The letter at the
point of intersection (vertex) goes in the middle.
K
H
J
❏ Write down the names of all the angles in this diagram.
I
................................................
................................................
................................................
................................................
................................................
................................................
Page 10
MEASURING ANGLES 2
❏ 1. Measure angle ∠ABC
using a protractor.
∠ABC = ___________
❏ 2. Look at the angles in the diagrams below.
❏ Find the size of these angles.
∠GHI =
∠LMN =
∠QRS =
∠QSR =
❏ 3. Measure the angles listed under each triangle.
C
R
A
∠JFB = ......................................
∠CAR = ....................................
∠XYZ = ......................................
∠FBJ = ......................................
∠RCA = ....................................
∠YXZ = ......................................
∠FJB = ......................................
∠ARC = ....................................
∠YZX = ......................................
❏ 4. Measure the angles in the diagrams below.
∠TDP = ..............................∠WDV = ....................∠GEK = .........................∠QOG = ....................
Page 11
ANGLES IN A TRIANGLE
❏ 1. Measure each of the angles in these triangles. Add them together and write the total in the space below.
c
a
a = ..............
a = ......................
b
b = ..............
b
b = ......................
+ c = ______
__________
+ c = __________
a
c
__________
What did you find out? ....................................................................................................................................
❏ 2. Calculate the angle size of the triangles below.
What do the angles of each of the triangles add up to? ..................................................
Rule: The sum of the angles of any
triangle is ............................˚
❏ 3. Using the rule, find the size of the angle x below without using your protractor.
❏ 4. Solve the following problems by using the clues.
This triangle is an
equilateral triangle
and all angles
are the same size.
∠WJA is a right angle
∠JAW = 47˚
∠CRZ =
Page 12
∠AWJ =
ANGLE FACTS
❏ Cut out four paper triangles on a spare piece of paper, based on these examples:
Right
Angle
Isosceles
Scalene
Equilateral
❏ Label the angles of each triangle.
❏ Carefully tear the corners off the right angle triangle and arrange them in a line like so:
❏ Repeat this process with the other three triangles and paste their angles below.
Isosceles
Scalene
Equilateral
What did you find? ..........................................................................................................................................
Try this activity with two different sized quadrilaterals.
❏ Paste the angles in the space below.
What did you find? ..........................................................................................................................................
Page 13
SCALENE TRIANGLES
A scalene triangle is a triangle that has three sides of different lengths. It also has no equal angles.
❏ Measure the angles in these three triangles.
Write the angle sizes in the correct place in each triangle.
❏ Draw three scalene triangles using a ruler
and then measure each of the angles using a protractor.
❏ Circle the scalene triangles below.
a.
b.
e.
f.
c.
d.
g.
❏ List some objects in your classroom or playground that include a scalene triangle.
For example, the slide in the playground.
......................................................................................
......................................................................................
......................................................................................
......................................................................................
Page 14
ISOSCELES TRIANGLES
An isosceles triangle has two equal sides. Measure the angles in each of the isosceles triangles below.
❏ 1. Write a rule about what you found.
..........................................................................................................................................................................
..........................................................................................................................................................................
❏ 2. Without using a protractor, find the missing angles in these isosceles triangles.
a.
b.
c.
❏ 3. Calculate the missing angles in the symmetrical house below without using a protractor. Use the other
rules you know of to work out the angles.
s = ..............................
m = ..............................
a = ..............................
r = ..............................
t = ..............................
i = ..............................
e = ..............................
Page 15
EQUILATERAL TRIANGLES
Equilateral triangles have three sides of equal length and angles of the same size.
Without measuring the equilaterals below, what size must each of the angles be? ..........................................
Equilateral triangles are always the same shape, just different sizes.
❏ Measure the angles in the triangles below to see if they are equilaterals and tick the ones that are. You will
have to be spot on with the protractor as some of them are pretty close!
a.
d.
b.
c.
e.
f.
❏ Construct an equilateral triangle.
Draw a line in the space below.
Using a compass place the point on the left end of the line and the pencil exactly on the right end of the
line.
Carefully mark a small line by swinging the compass around.
Make sure you do not change the width at which the compass is set.
Now repeat this action by placing the point of the compass on the other
end of the line.
Join the ends of the line to the intersection of the two arcs (curves).
Measure the angle sizes using a protractor.
Page 16
A right triangle is a triangle that has one angle that measures exactly 90˚.
It is often marked with a small square in the right angle.
hyp
ote
nu
se
GET THE RIGHT ANGLE
❏ Mark an X in the right triangles below. You may like to use a protractor to double check.
The side of the right triangle that is directly opposite the right angle
is known as the hypotenuse.
❏ Label the hypotenuse on each right triangle above and also the triangles below.
Pythagoras
The famous Greek mathematician, Pythagoras,
made an important discovery about right triangles.
He found that, in any right triangle, the square of
the hypotenuse is equal to the sum of the squares
of the other two sides.
❏ Find the length of the hypotenuse
in each of these triangles
using this formula.
132 = 52 + 122
169 = 25 + 144
b.
a.
❏ Challenge: Measure the length of sides a and b of this
right triangle. Using a calculator, check to see what the
hypotenuse should be using Pythagoras’ discovery.
Use your ruler to check if this is correct.
Try to be exact. Show the figures in this space.
c2 = a2 + b2
Round your answer to two decimal places.
c.
Page 17
ANGLING AROUND
So far we know that:
All angles in a triangle add up to 180˚.
The sum of any angles that make up a straight line is 180˚.
All angles in a quadrilateral add up to 360˚.
❏ Use these rules to find the missing angle x in the diagrams below.
1.
2.
x= ........................
x= ........................
3.
4.
x= ........................
x= ........................
5.
6.
O
∠MNK = 58˚
∠ONL = 37˚
∠FLP = 109˚
∠LFE = 51˚
x = ......................
x = ....................
7.
8.
x = ......................
x = ......................
Page 18
INTERSECTING LINES
❏ Measure the angles in the diagram below.
∠ACD = ..........................................
∠BCE = ..........................................
∠DCE = ..........................................
∠ACB = ..........................................
❏ Complete this sentence to make up a new rule.
When two straight lines intersect .............................................................................................................
❏ Use this rule to find the size of angle x below.
1.
2.
3.
4.
5.
6.
x=
1....................... 2 ...................... 3....................... 4....................... 5....................... 6.......................
Challenge: Find the values of all the other angles in each diagram above and write them in.
Remember the rule: Angles along a straight line add up to 180˚.
Page 19
PARALLEL LINES
In the diagrams below measure the size of angles x and y. Are they congruent?
Try out some more examples of your own on the back of this page.
❏ Find the size of these angles by using all the rules that you know.
Remember: Angles made by intersecting lines are congruent.
You do not need a protractor for this activity.
z
w = ............˚
zz = ............˚
x = ............˚
y = ............˚
b = ............˚
c = ............˚
d = ............˚
❏ Measure the angles below.
a = ............˚
What did you find? ..........................................................................................................................................
Page 20
DEGREES IN A CIRCLE
1. ❏ Using a protractor, measure the angles in each of the circles below.
❏ Add the angle sizes for each circle.
w = ..............˚
d = ..............˚
l = ..............˚
x = ..............˚
e = ..............˚
m = ..............˚
y = ..............˚
f = ..............˚
n = ..............˚
z = ..............˚
g = ..............˚
k = ..............˚
h = ..............˚
j = ..............˚
Total .............
Total .............
Total .............
...................˚
...................˚
...................˚
❏ Write a rule about what you found.
..........................................................................................................................................................................
..........................................................................................................................................................................
2. ❏ Without using a protractor, find the missing angles below using the rules you know. Then check that
all the angles add up to 360˚.
∠f = .................
∠g = .................
∠v = .................
∠j = .................
∠c = .................
∠r = .................
Remember:
Angles that exist along a line are supplementary angles and always add up to 180˚.
Page 21
CONSTRUCTING ANGLES 1
Angles can be constructed by using a protractor.
To make an angle of 53˚, place your protractor on the line PR below so that the middle point is at the end of the line
(P) and the 0˚ mark points to R. Carefully find 53˚ on your protractor using the scale that starts at “0” and mark the
point at the end. Rule a line to this point.
1. ❏ Construct the following angles using this method.
23˚
81˚
45˚
72˚
These angles are all .................................................................. because they are less than ..............................˚
112˚
135˚
178˚
95˚
These angles are all .................................................................. because they are more than ..........................˚
2. ❏ From the point in the lines below draw a line to construct an angle of the value underneath the line.
Write the value of the other angle you have made.
These angles are known as ............................................ angles because they add up to ................................˚
39˚
Page 22
157˚
116˚
CONSTRUCTING ANGLES 2
It is possible to construct some angles without the use of a protractor. By carefully using a compass we can
make exact angles using simple rules. The easiest angle to construct is probably a 60˚ angle.
❏ On the line JK below, place the compass point at the J and stretch the pencil end to K. Without changing
the width, draw an arc. Repeat this step using K as the compass point. Where these lines intersect draw a
line. This angle should now measure exactly 60˚.
To make a 30˚ angle we can bisect this 60˚ angle, again using only a compass. To bisect an angle or a line
means to divide it into two equal sections.
Note: Make sure you do not
change the width of the compass
setting in your working.
arc
3. Where these arcs intersect draw
a line to the vertex.
b
4. Measure the two new angles to
check.
a
1. Place the compass point on the vertex and
draw an arc over the two rays.
2. Where the arc intersects the rays a and b, place the
compass point to draw two new arcs.
❏ Bisect this right angle to make a 45˚ angle using the method above.
❏ Measure the angle below. Bisect this angle using your compass.
What are the sizes of the new angles?
original angle = ............˚
new angles = ............˚
Page 23
ANGLES AND DIRECTIONS 1
❏ Look at the path in the grid. Follow the path and read the directions as you go.
The path starts at X and heads north for two squares.
A turn was then made to the left at a 90˚ angle and then
after moving one square, another turn was made at a 90˚
angle to the right also moving one square. The next turn
was a 45˚ turn to the left and moving along two squares.
A 45˚ turn to the left was then made and continued for only
1 square. A 90˚ turn to the left was made and the path moved
5 squares. Another 90˚ turn to the left was made and
continued 2 squares to the end of the path.
Find the treasure!
❏ Now that you’re an expert in following directions find where the treasure is buried below.
Start at the X and move right for 3 cm and then turn 90˚ to the left. Continue for 4 cm and then make a 90˚
turn to the right and continue for 5 cm. Turn 45˚ to the right and travel 1 cm. Now turn 45˚ to the right
and move 1 cm. Make a 90˚ turn to the right and move 8 cm. Again turn 90˚ to the right and move 3 cm.
Turn left at a 90˚ angle and move 1 cm. Mark another X where the treasure is buried.
Describe in words the path you would take from your classroom to the library giving the directions and
approximate angles. For example, head south for 30 meters and make a right turn at the office. Walk for 20
meters and then make a 45˚ turn to the left, etc.
..........................................................................................................................................................................
..........................................................................................................................................................................
..........................................................................................................................................................................
..........................................................................................................................................................................
..........................................................................................................................................................................
❏ Check your description with that of one of your friends and then draw a map on a scrap piece of paper
based on your description.
Page 24
ANGLES AND DIRECTIONS 2
The compass below shows the angles of each direction.
❏ Label the following points on the compass.
North(N), South(S), East(E), West(W),
Northwest (NW), Southwest (SW),
Northeast (NE), Southeast (SE).
❏ Follow the paths given using the grids below. Only move one square at a time.
1. NW, N NW, W, N, E, NE, N, N, SE, NE, S, S, SE, E, S, W, SW, S, SW.
Start at the X.
1.
2.
2.
N, E, N, W, N, E, E, N, W, N, NE, NW,
NW, NE, SE, NE, SE, NE, S, S, W, W, S, E, E,
S, SW, NW, S, SE, S, W, S, E, S, W, W, W, W.
❏ On a piece of 1 cm grid paper make a pattern and
then give instructions for a friend to draw the pattern.
Page 25
BILLIARDS ANGLES
Below is a diagram of a billiards or pool table with only four pockets. The way it works is very simple. If you
hit the ball it then bounces off the side cushion at a 90˚ angle until it finally lands in a pocket. In the diagram below, the track of the ball is mapped out for you.
Pocket
❏ Using this method, work out which pocket each of the following balls will go into if they are hit in the
direction of the arrow. Draw a circle around the pocket.
a.
b.
c.
d.
e.
f.
Page 26
ANGLE CHECK POINT
❏ How many angles can you find? Use all the rules you have learned so far to find the size of the missing
angles in each diagram below. You should only need to use a protractor to check if you are correct.
a.
b.
c.
x
y
y
d.
x
x
X
e.
f.
g.
h.
z
x
x
y
x
x
i.
j.
y
k.
l.
x
x
m.
x
x
x
n.
y
y
x
z
o.
y
x
❏ Complete the following sentences.
1.
Supplementary angles add up to ..........................˚
2.
Angles in an equilateral triangle are .....................................................................
3.
Opposite angles formed by two intersecting lines are ...............................................................
4.
Angles in a circle add up to ......................˚.
5.
Angles in a quadrilateral add up to ..........................˚.
6.
How many degrees are there in a straight line? ..............................˚
7.
A right triangle is a triangle that has .....................................................................................................
8.
A reflex angle is .....................................................................................................................................
9.
Complementary angles add up to ........................˚
Page 27
BASEBALL HITS
Look at the baseball pitch below. The dots in the diamond indicate where the ball has landed. You may need
to draw in some guide lines to help you measure.
X batter
6
8˚
7
8
2
5
3
4
❏ Find the angle of the shot for each of the hits above. The first one has been done for you.
8°
1 = ................................
2 = ................................
3 = ................................
4 = ................................
5 = ................................
6 = ................................
7 = ................................
8 = ................................
Page 28
AN ANGLE ON TIME
We see changing angles on a clock face every minute.
What sized angle is shown for the times below?
Hint: Use this clock face and pencil lines to help you.
5:30 - 15˚
11:00 ..............
Noon ..............
3:00 ................
7:00 ................
8:30 ................
9:00 ................
3:45 ................
e.g.
❏ Draw the times in the clock faces below using your ruler.
What angles are they showing?
7:16 ................˚
11:17 ................˚
Clock Quiz
1. How many degrees does an hour hand move between 11:00 and 12:00? ......................
2. How many degrees are formed at exactly 6:00? ......................
3. After 3:00, when is the very next time that the hands form a perfect right angle? ......................
4. What is the reflex angle size when a clock’s hands say 4:30? ......................
5. What kind of angle is formed by the clock hands at these times? Measure from the hour hand
................................................................................................
e.x. 11:45 - reflex
a. 3:45 ....................................
b. 11:30 ..................................
c. 9:33 ....................................
d. 7:45 ....................................
e. 12:45 ..................................
f. 2:17 ....................................
g. 1:55 ....................................
h. 5:58 ....................................
i. 12:37 ..................................
Page 29
ANGLES IN THE REAL WORLD
(Hint : Draw diagrams where necessary.)
1. Mark is facing south. If he turns 90˚ to the right, what direction will he be facing?
2. Irene owns a triangular block of land. She knows that one corner is a 45˚ angle and the other corner is a
right angle. What is the angle size of the third corner?
3. Katie was heading toward the markets in an eastern direction. She made a 90˚ turn to her right after
100 meters and then another 90˚ turn to her right about 200 meters further. In what direction was she
now heading?
4. Lesley noticed that the hands on the clock made a perfect 90˚ angle and that one of the hands was on
the three. What time must it have been?
5. Denis walked 300 meters in a northern direction and made a 90˚ turn to the right. He then walked a
further 250 meters and then made a turn south. How many degrees did he turn and in which direction?
6. Sarah was putting up a sail on her yacht. She noticed the sail was an isosceles triangle and that the smallest corner was a 30˚ angle. What must the sizes of the two other angles be?
7. The council is planning to build a new slide in the park. The ladder, which stands at 3 meters high when
at right angles to the ground, will have a long slide attached to the top. The end of the slide is 7 meters
from the bottom of the ladder.
What angle will the slide be from the ground
based on these measurements? Use the box
below, a ruler and a pencil to help you draw the
model of the slide and then use a protractor to
find the size of the angle.
1 box = 1 meter.
Angles for dinner?
Emily made a pizza for the family. She knew Carl would want the
biggest piece and the size of the angle of his slice was 80˚.
There were four other people that needed a piece.
If she made them exactly the same size what angle
would each slice be at the point?
For dessert, a fresh apple pie was
to be cut into 5 equal pieces.
What size angle should each piece be at the
point if they are cut to exactly the same size? ....................................
Page 30
PUZZLES WITH ANGLES
❏ Use all the rules about angles and lines to help you solve the puzzles below.
Note: You do not need a protractor to solve the puzzles.
Lines on this page that look parallel are parallel and angles which look 90˚ are 90˚.
❏ Find the size of every angle in the gates below. Only one angle is given.
Hint: Remember, isosceles triangles have two equal angles.
Page 31
PARTS OF A CIRCLE 1
In the circle below we have:
a chord - CG
a radius - AE
a diameter - BF
radii - DE and BE
Note: Radii is the plural of radius.
In the circle below draw:
A chord - YZ
A radius - XW
A diameter - RS
Other radii - QW and TW
Measure the lengths of the radii in this circle
and write them below.
XW = ..........................
QW = ..........................
TW = ..........................
Measure the diameter RS ..............................
What do you notice?........................................
In the circles below, name the radii and find their lengths.
M
J
Z
MG - ......................................
JB -..........................................
ZX - ........................................
MI - ........................................
JF - ..........................................
ZY - ........................................
MR -........................................
JC -..........................................
ZW - ......................................
Page 32
PARTS OF A CIRCLE 2
❏ Using a compass, ruler and a pencil draw circles with the following radii around the points below.
.
.
.
radius = 2 cm
radius = 3.5 cm
radius = 1.5 cm
Arcs
An arc is a section of a curve with two end points.
❏ Look at the chord CD.
The chord divides the circle into two arcs, the major
arc which is longer and the minor arc which is shorter.
D
C
Arc CSD is the __________________ arc
and Arc CTD is the __________________ arc.
❏ Label the minor and major arcs of the circles below.
The first one has been done for you.
major arc = JEN
major arc = ..................................
minor arc = JAN
minor arc = ..................................
major arc = ..............................
minor arc =..............................
major arc = ..............................
minor arc =..............................
major arc = ..............................
minor arc =..............................
Page 33
TRIANGLES IN CIRCLES 1
❏ Join a line to the points to make the following triangles in the circles:
1. Triangle GQR
2. Triangle WTH
3. Triangle SJP
4. Triangle FMK
Is the triangle in the minor or major arc of the circle?
1. ..................................
2. ..................................
3. ..................................
4. ..................................
❏ Using a protractor find the size of each angle in the triangles.
∠G= ............................
∠W= ............................
∠S= ..............................
∠F = ............................
∠Q = ............................
∠T = ............................
∠J= ..............................
∠M = ............................
∠R = ............................
∠H = ............................
∠P= ..............................
∠K = ............................
What do the angles of each triangle total? ..................................................
❏ In the circles below draw and label the stated triangles and measure the angles.
ΔABC
∠A = ..................˚
ΔXYZ
∠B = ..................˚
∠C = ..................˚
∠X = ..................˚
ΔQRS
∠Y = ..................˚
∠Z = ..................˚
∠Q = ..................˚
∠R = ..................˚
∠S = ..................˚
Page 34
TRIANGLES IN CIRCLES 2
❏ In the triangles below, name the arc in which the triangle has been drawn
(major or minor) and decide whether the angle x is an acute, obtuse or right angle.
1. arc - ..................................
angle -..............................
2. arc - ..............................
x
angle -............................
x
x
3. arc - ..............................
x
angle - ..........................
4. arc - ................................
angle -............................
x
5. arc - ................................
angle -............................
6. arc - ................................
angle -............................
x
7. arc - ................................
x
angle -............................
8. arc - ................................
x
angle -............................
What did you find?
Angles formed in the minor arc using the end points of a chord are: ..............................................................
..........................................................................................................................................................................
Angles formed in the major arc using the end points of a chord are: ................................................................
..........................................................................................................................................................................
Angles formed in a semicircle using the end points of the diameter are: ..........................................................
..........................................................................................................................................................................
Page 35
ANGLES IN CIRCLES 1
FI is a chord in the circle H.
Measure angles FGI and FHI
using a protractor.
∠FGI = ..................................˚
∠FHI = ..................................˚
What did you find?
In the circle K draw a chord LM.
Draw a triangle in the major arc and label it LNM.
Construct triangle KLM.
Measure these angles:
∠LNM = ......................˚
∠LKM = ......................˚
❏ Make up a rule to describe what
you found in both of the circles above.
..........................................................................................................................................................................
..........................................................................................................................................................................
❏ In each circle below construct a triangle
using the chord and the point in the major arc of the circle.
Z
C
T
A
B
A
B
A
B
❏ Measure the angle on the perimeter in each circle. What did you find?
..........................................................................................................................................................................
..........................................................................................................................................................................
Page 36
ANGLES IN CIRCLES 2
❏ In the circle D join the points to make triangle CDE.
❏ Make the triangle CFE in the minor arc.
❏ Measure the angles below using a protractor:
∠CDE = ________˚
∠CFE = ________˚
What did you find? ..............................................................................
................................................................................................................
❏ In each circle below draw a chord and then construct two triangles - one using the chord and any point in
the minor arc and one using the chord and the center point. Label your diagrams.
❏ Measure the two angles in each of the circles using your protractor.
What kind of angle is the angle at the perimeter in each circle?
❏ Make up a rule to describe what you found.
..........................................................................................................................................................................
..........................................................................................................................................................................
❏ Without using a protractor, find the size of x in each circle below.
Page 37
SHAPES IN CIRCLES 1
Hexagons
❏ Place the point of your compass in the center.
Set your compass so that the pencil touches the
perimeter and then move the compass point to
any point on the perimeter.
Mark off points on the perimeter until you
have six marks evenly spaced around the circle.
Rule a line from mark to mark to form the hexagon.
Measure the angles of the hexagon. ..............................
Are the angles congruent? ...........................
Squares
❏ Using your compass, place the point on one
end of the diameter and make an arc at the
top of the circle and the bottom of the circle.
Repeat this by placing the point at the other
side.
Join the two points where the arcs intersect
to make a second diameter of the circle.
Now join the ends of the diameter in the circle to form the square.
❏ Measure each angle in the square. If you were
exact with the construction, each angle
should be ..............................˚
❏ Using a compass, construct some symmetrical shapes or patterns in the circles below.
Page 38
SHAPES IN CIRCLES 2
❏ Measure each of the angles in the circle below using a protractor.
What size is each angle? ............................˚
Multiply this number by the number of angles
(5).
............................˚ x 5 = ..........................˚
Join up the points where the radii meet the
edge of the circle. What shape have you
formed?
....................................................................
Knowing that all angles in circles add up to 360˚, construct the following regular shapes in the circles below
by first working out how many degrees there are in each angle of the shape. You will need to be exact with
the angles so that the shapes are regular.
Note: you do not need a compass for this activity.
Hexagon
Equilateral Triangle
Octagon
Nonagon (9 sides)
Decagon (10 sides)
Dodecagon (12 sides)
Page 39
ANSWERS
Looking at Different Angles (page 5)
Right angles - 4, 6, 9; obtuse angles - 3, 5, 7; acute angles - 1, 2, 8.
Measuring Angles 1 (page 6)
3. 20˚ a - 63˚, acute
b - 30˚, obtuse
f - 165˚, obtuse
g - 28˚, acute
Reflex Angles (page 7)
1.
a = 335˚
360˚ angles
a = 65˚
b = 240˚
b = 63˚
c - 26˚, acute
h - 90˚, right
c = 340˚
c = 110˚
d = 60˚
d = 70˚
d - 90˚, right
i - 120˚, obtuse.
e = 70˚
e = 52˚
e - 60˚, acute
f = 55˚
x = 80˚
Angles on the Line (page 8)
Supplementary angles - 90˚, 30˚, 55˚.
Complementary angles - 33˚, 45˚, 6˚, 27˚.
102˚, 58˚, 45˚.
Which Angle is Larger? (page 9)
Note: Please note angles may vary slightly due to the printing process.
These answers should be used as a guide.
1) A = 47˚, C = 56˚
2) O = 109˚, K = 94˚
3) W = 16˚, N = 25˚
4) G = 133˚, S = 120˚
5) R = 110˚, Q = 70˚
6) Z = 60˚, U = 139˚
8) N = 71˚, P = 45˚
9) T = 90˚, K = 87˚.
Congruent.
7) E = 80˚, V = 60˚
Naming Angles (page 10)
Answers will vary. ∠FHI, ∠FHG, ∠GHJ, ∠HGJ, ∠GJM, ∠MJK.
Measuring Angles 2 (page 11)
1.
∠ABC = 42˚
2. ∠GHI = 135˚
∠LMN = 65˚ ∠QRS = 62˚
3.
∠JFB = 60˚ ∠FBJ = 60˚
∠CAR = 105˚ ∠RCA = 45˚
∠XYZ = 65˚ ∠ = 43˚
4.
∠TDP = 80˚ ∠WDV = 180˚∠GEK = 20˚ ∠QOG = 45˚.
∠QSR = 64˚.
∠FJB = 60˚
∠ARC = 30˚
∠YZX = 72˚
Angles in a Triangle (page 12)
1.
a = 25˚ b = 130˚
c = 25˚.
2.
They should all add up to 180˚
3.
x = 53˚ x= 39˚ x = 68˚ x = 21˚.
4.
∠CRZ = 60˚ ∠AWJ = 43˚.
a = 60˚ b = 45˚ c = 75˚.
Angle Facts (page 13)
Checks diagrams on page.
Scalene Triangles (page 14)
Check diagrams of triangles. Scalene - b, f, g.
Isosceles Triangles (page 15)
1.
An isosceles triangle always has two congruent angles at the base of the two congruent sides.
2.
a) g - 81˚, m - 18
b) g - 68˚, m - 44
3.
s - 130˚, m - 25˚, a - 58˚, r - 58˚, t - 32˚, i - 32˚, e - 58˚
Equilateral Triangles (page 16)
60˚. Equilateral triangles - a, e.
Page 40
c) p - 75˚, q - 30˚
Get the Right Angle (page 17)
Check diagrams on sheet. Pythagoras - a. 10 cm, b - 5 cm, c - a= 2 cm, b = 6 cm, c ∪ 6.32.
Angling Around (page 18)
1) x=123˚
2) x=69˚
3) x=133˚
4) x=27˚
5) x=85˚
6) x=58˚
7) x=40˚
8) x=30˚
Intersecting Lines (page 19)
∠ACD = 30˚ ∠BCE = 30˚ ∠DCE = 150˚ ∠ACB = 150˚. The opposite angles are equal in size.
1. x=125˚
2. x=37˚
3. x=117˚
4. x=45˚
5. x=94˚
6. x=90˚
Parallel Lines (page 20)
x and y are congruent.
w = 105˚ z = 105˚
x = 110˚ y = 110˚
a = 45˚ b = 135˚
c = 150˚ y = 30˚
Each pair of angles within the parallel lines adds up to 180˚.
Degrees in a Circle (page 21)
1.
w - 90˚ x - 90˚ y - 90˚ z - 90˚ d - 45˚ e - 45˚ f - 90˚
g - 135˚
h - 45˚ l - 45˚ m - 105˚
n - 30˚ k - 45˚ j - 135˚
2.
f = 125˚
A circle has 360˚.
j = 65˚ g - 180˚
c - 55˚ v = 46˚ r = 115˚
Constructing Angles 1 (page 22)
1. Check student’s diagrams. 2. Supplementary, 180˚.
The angles should measure 141˚, 23˚, 64˚.
Constructing Angles 2 (page 23)
Check diagrams.
Angles and Directions 1 (page 24)
Angles and Directions 2 (page 25)
Billiards Angles (page 26)
a) lower left-hand pocket, b) top right-hand pocket, c) top right-hand pocket,
d) lower left-hand pocket, e) top left-hand pocket, f) top left-hand pocket.
Page 41
Angle Check Point (page 27)
a) x=50˚ b) x=80˚, y=80˚ c) x=130˚, y=50˚ d) x=60˚ e) x=90˚ f) x=120˚
g) x=85˚, y=95˚, z=95˚ h) x=335˚ i) x=309˚ j) x=55˚ k) x=70˚, y=40˚
l) x=25˚ m) x=120˚, y=60˚, z=60˚ n) x=87˚, y=87˚, z=93˚ o) x=145˚, y=35˚
1) 180˚
2) congruent
3) congruent
4) 360˚
5) 360˚
6) 180˚
7) an angle of exactly 90˚
8) an angle between 180 and 360˚
9) 90˚
Baseball Hits (page 28)
Note: - Answers are approximate - 1) 8˚ 2) 31˚ 3) 16˚ 4) 43˚ 5) 61˚ 6) 66˚ 7) 97˚ 8) 46˚
An Angle on Time (page 29)
Answers may vary, check diagrams.
Clock Quiz
1) 30˚
2) 180˚
3) 3.33
4) 315˚
5) a) obtuse b) obtuse c) reflex d) acute e) reflex f) acute g) reflex h) obtuse i) reflex.
Angles in the Real World (page 30)
1) West2) 45˚ 3) West4) 3.00 5) He turned right 90˚ 5) Both angles are 75˚
7) Check diagram - angle should be 25˚.
The slide angle will be 23˚.
Angles for Dinner? - pizza -70˚, pie -72˚.
Puzzles With Angles (page 31)
Check diagrams.
Parts of a Circle 1 (page 32)
Check diagram. All radii will be 4 cm. Diameter = 8 cm.
Circle M - all radii are 1.5 cm, Circle J - all radii are 2.3 cm, Circle Z - all radii are 2 cm.
Parts of a Circle 2 (page 33)
Check circle diagrams. Arcs - Arc CSD is the minor arc and arc CTD is the major arc. Answers will vary
depending on labelling.
Triangles in Circles 1 (page 34)
1. major, 2. major, 3. minor, 4. major.
Answers will vary depending on where students connect to points. The answers below are approximate.
∠G= 40˚
∠W= 50˚
∠S= 50˚
∠F = 70˚
∠Q = 80˚
∠T = 80˚
∠J= 100˚
∠M = 50˚
∠R = 60˚
∠H = 50˚
∠P= 30˚
∠K = 60˚
They all total 180˚ .
Check diagrams and angles.
Triangles in Circles 2 (page 35)
1. major arc, acute; 2. minor arc, obtuse; 3. neither, right angle; 4. minor, obtuse; 5. neither, right angle, 6.
major, acute; 7. major, acute, 8. neither, right angle.
Students should find that angles formed in the minor arc are obtuse, angles formed in the major arc are acute
and that all angles formed using the end points of the diameter are right angles.
Page 42
Angles in Circles 1 (page 36)
∠FGI = 60˚ ∠FHI = 120˚.
Answers will vary according to diagrams. The outer angle is half the size of the central angle.
Angles in Circles 2 (page 37)
Answers may vary according to drawings. The angles CDE and CFE should be congruent.
x = 70˚, x = 60˚, x = 70˚, x = 45˚.
Shapes in Circles 1 (page 38)
Hexagons - the angles should be congruent; Squares - each angle should be 90˚.
Shapes in Circles 2 (page 39)
72˚, 72 X 5 = 360, pentagon.
Page 43
P
r:
c
t
e
f
f
o
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In-class review
Tutoring
Home-schooling
GRADES 5-6
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